Proof. Let $\sigma$ be the number of Laplacian eigenvalues of $T_1\cup T_2$ which are greater than or equal to the average vertex degree $\overline{d}(T_1\cup T_2)=2-\frac{4}{n}$, and let $n_1+n_2=n$. Let $k_i$ be the number of Laplacian eigenvalues of $T_i$ which are greater than or equal to average vertex degree $\overline{d}(T_1\cup T_2)=2-\frac{4}{n}$. Since $n_1\ge n_2$ implies that $\overline{d}(T_1)\ge \overline{d}(T_1\cup T_2)$ and $\overline{d}(T_2)\le \overline{d}(T_1\cup T_2)$, it follows that $k_1\ge \sigma_1$. Since $\mu_{\sigma_1+1}- \overline{d}(T_1)<\frac{-2}{n}$, therefore

\begin{equation*}
\begin{split}
\mu_{\sigma_1+1}- \overline{d}(T_1\cup T_2)&=\mu_{\sigma_1+1}-2+\frac{4}{n}\\&=\mu_{\sigma_1+1}-2+\frac{2}{n_1}+\frac{4}{n}-\frac{2}{n_1}\\&
\end{split}
\end{equation*}

as $n>n_1$. Thus, it follows that $k_1\le \sigma_1$ and so we must have $\sigma_1=k_1$. Now, proceeding similar to Theorem <ref> the result follows.

§ CONCLUSION

In order to prove the conjecture in general, one must be aware of the distribution of Laplacian eigenvalues of trees around the average degree. The graph invariant $ \sigma $ investigated in [41], plays a fundamental role in finding the lower bounds of  $ S_{k}(G) $, which in turn may help in proving the Laplacian energy conjecture. Thus to prove the Laplacian energy conjecture, we must study $ \sigma $ of the trees and use the  gained information in verifying the Laplacian energy conjecture. Some tree transformation as in [137] can also help in verifying the Laplacian energy conjecture.

CHAPTER: NORMALIZED LAPLACIAN EIGENVALUES AND EIGENVALUES OF POWER GRAPHS OF FINITE CYCLIC GROUPS

In this chapter, we discuss the normalized Laplacian eigenvalues of the joined union of regular graphs in terms of the adjacency eigenvalues and the eigenvalues of the quotient matrix associated with graph $ G $. As a consequence of the joined union of graphs, we investigate the normalized Laplacian eigenvalues of the power graphs of finite cyclic group $ \mathbb{Z}_{n}. $

§ INTRODUCTION

The normalized Laplacian matrix of a graph $ G $, denoted by $ \mathcal{L}(G) $, whose rows and columns are indexed by the vertices of $ G $, is defined as

\begin{equation*}
\mathcal{L}(G)=\begin{cases}
1 & \text{if} ~ v_{i}=v_{j} ~\text{and} ~ d_{v_{i}}\neq 0,\\
\dfrac{-1}{\sqrt{d_{v_{i}}d_{v_{j}}}} & \text{if}~ v_{i}\sim v_{j},\\
0 & \text{otherwise}.

\end{cases}
\end{equation*}

This matrix was introduced by Chung [29] to study the random walks of $ G $. This matrix is equivalently defined as

\begin{align*}
\mathcal{L}(G)=& D(G)^{-\frac{1}{2}}L(G)D(G)^{-\frac{1}{2}}\\
=& D(G)^{-\frac{1}{2}}(D(G)-A(G))D(G)^{-\frac{1}{2}}\\
=& I-D(G)^{-\frac{1}{2}}A(G)D(G)^{-\frac{1}{2}},
\end{align*}

where $ D(G)^{-\frac{1}{2}} $ is the diagonal matrix whose $ i$-th diagonal entry is $ \frac{1}{\sqrt{d_{i}}} $. Clearly, $ \mathcal{L}(G) $ is real symmetric and positive semi-definite matrix so its eigenvalues are real, ordered as $ 0=\rho_{n}(\mathcal{L}) \leq \rho_{n-1}(\mathcal{L})\leq \dots \leq \rho_{1}(\mathcal{L})$ and are known as normalized Laplacian eigenvalues of $ G $. We see that $ D^{-\frac{1}{2}} \textbf{J}_{n\times 1} $ is an eigenvector of $ \mathcal{L}(G) $ with corresponding eigenvalue $ 0 $, where $ \textbf{J} $ is a vector with each entry $ 1 $. In [29], it is shown that $ \sum\limits_{i=1}^{n} \rho_{i}(\mathcal{L})=n $, for each $ 1\leq i\leq n $, $~ \rho_{i}(\mathcal{L})\in [0,2] $ and $  \rho_{n}(\mathcal{L})= 2 $ if and only if a connected component of $ G $ is bipartite. Cavers et. al. [21] studied the normalized Laplacian energy and characterized the extremal graphs. Two non-isomorphic graphs $ G_{1} $ and $ G_{2} $ are normalized Laplacian cospectral if $ G_{1} $ and $ G_{2} $ have same normalized Laplacian eigenvalues. Butler [18] constructed normalized Laplacian cospectral graphs using twins and scaling technique. More work on $\mathcal{L}(G)$ can be seen in [20, 39, 43, 142] and references therein.

§ NORMALIZED LAPLACIAN EIGENVALUES OF THE JOINED UNION OF GRAPHS

Consider the matrix

\begin{equation*}
M= \begin{pmatrix}
m_{1,1} & m_{1,2} & \cdots & m_{1,s} \\
m_{2,1} & m_{2,2} & \cdots & m_{2,s} \\
\vdots & \vdots & \ddots & \vdots \\
m_{s,1} & m_{s,2} & \cdots & m_{s,s} \\
\end{pmatrix}_{n\times n}, \end{equation*}

whose rows and columns are partitioned according to a partition $P=\{ P_{1}, P_{2},\dots , P_{m}\} $ of the set $X= \{1,2,\dots,n\}. $ The quotient matrix $ \mathcal{Q} $ of the matrix $ M $ is the $s \times s$ matrix whose entries are the average row sums of the blocks $ m_{i,j} $. The partition $ P $ is said to be  equitable if each block $ m_{i,j} $ of $ M $ has constant row (and column) sum and in this case the matrix $ \mathcal{Q} $ is called as  equitable quotient matrix. In general, the eigenvalues of $ \mathcal{Q}  $ interlace the eigenvalues of $ M $. In case the partition is equitable, we have following lemma.

If the partition $ P $ of $ X $ of matrix $ M $ is equitable, then each eigenvalue of $ \mathcal{Q} $ is an eigenvalue of $ M. $

If each of the $ G_{i} $ is $ r_{i} $ regular, then the following result gives the normalized Laplacian eigenvalues of $G[G_1,G_2,\dots,G_n]$ in terms of the adjacency eigenvalues of the graphs $G_1,G_2,\dots,G_n$ and the eigenvalues of the quotient matrix.

[fill=black] (0,0) circle (2pt);

at (0,-.3) $ G_{1} $;

[fill=black] (0,0) circle (2pt);[fill=black] (0,1) circle (2pt);
[thin] (0,0)–(0,1);

at (0,-.3) $ G_{2} $;

[fill=black] (0,0) circle (2pt);[fill=black] (1,0) circle (2pt);[fill=black] (0.5,0.5) circle (2pt);

[thin] (0,0)–(1,0); [thin] (0,0)–(0.5,0.5);[thin] (1,0)–(0.5,0.5);

at (.5,-.3) $ G_{3} $;

[fill=black] (-1,0) circle (2pt);[fill=black] (2,0) circle (2pt);[fill=black] (5,0) circle (2pt);

[thin] (-1,0)–(2,0); [thin] (2,0)–(5,0);

at (-1,.3) $ v_{1} $;at (2,.3) $ v_{2} $;at (5,.3) $ v_{3} $;

at (2,-.3) $ G=P_{3} $;

[fill=black] (0,0) circle (2pt);[fill=black] (1,1) circle (2pt); [fill=black] (1,-1) circle (2pt);
[fill=black] (4,0) circle (2pt);[fill=black] (3,1) circle (2pt); [fill=black] (3,-1) circle (2pt);

[thin] (0,0)–(1,1); [thin] (0,0)–(1,-1);[thin] (1,1)–(1,-1);
[thin] (4,0)–(3,1); [thin] (4,0)–(3,-1);[thin] (3,1)–(3,-1);

[thin] (1,1)–(4,0);[thin] (1,1)–(3,1);[thin] (1,1)–(3,-1);
[thin] (1,-1)–(4,0);[thin] (1,-1)–(3,1);[thin] (1,-1)–(3,-1);

at (2,-1.5) $ P_{3}[G_{1},G_{2},G_{3}] $;

Joined union of three graphs

Let $G$ be a graph of order $n$ and size $m$. Let $ G_{i}$ be $r_{i}$ regular graphs of order $ n_{i} $ having adjacency eigenvalues $\lambda_{i1}=r_{i}\geq \lambda_{i2}\geq\ldots\geq \lambda_{in_{i}}, $ where $ i=1,2, \ldots, n$.  Then the normalized Laplacian eigenvalues of the graph $ G[G_{1},\ldots, G_{n}] $ consists of the eigenvalues $ 1-\dfrac{1}{r_{i}+\alpha_{i}}\lambda_{ik}(G_{i})$, for $ i=1,\ldots,n $ and $ k=2,3,\ldots, n_{i} $, where $\alpha_i=\sum\limits_{v_j\in N_{G}(v_{i})}n_{i}$ is the sum of the orders of the graphs $G_j, j\ne i$ which correspond to the neighbours of vertex $ v_{i}\in G $. The remaining $n$ eigenvalues are the eigenvalues of matrix (<ref>).

Let $V(G)=\{ v_{1}, \ldots, v_{n}\}$ be the vertex set of $ G $ and let $V(G_i)=\{ v_{i1}, \ldots, v_{in_i}\}$ be the vertex set of the graph $G_i$, for $i=1,2,\dots,n$. Let  $H=G[G_{1}, \ldots, G_{n}]$ be the joined union  of $ r_{i} $ regular graphs $G_i,$ for $ i=1,2,\dots,n $. It is clear that the order of $ H $ is $ N=\sum\limits_{i=1}^{n}n_{i} $. Since degree of each vertex $v_{ij}\in V(H)$ is degree inside $ G_{i} $ and  the sum of orders of $ G_{j}\text{'s}, j\ne i $, which corresponds to the neighbours of the vertex $ v_{i} $ in $ G $, where $1\le i\leq n$ and $1\leq j\leq n_i$, therefore, for  each $v_{ij}\in V(G_{i})$, we have

\begin{equation}\label{eq23}
d_{H}(v_{ij})=r_{i}+\sum\limits_{v_j\in N_{G}(v_i)}n_{j}=r_i+\alpha_i,
\end{equation}
where $\displaystyle\alpha_i=\sum\limits_{v_j\in N_{G}(v_i)}n_{j}$. Under suitable labelling of the vertices in $ H $, the normalized Laplacian matrix of $H$ can be written as

\begin{equation*}
\mathcal{L}(H)=\begin{pmatrix}
g_1 & a(v_1,v_2)& \ldots &  a(v_1,v_n) \\
a(v_2,v_1) & g_2 & \ldots &  a(v_2,v_n)\\
\vdots &\vdots &\ddots &\vdots\\
a(v_n,v_1) & a(v_n,v_2) &\ldots & g_n
\end{pmatrix},
\end{equation*}

where,  for $i=1,2,\ldots,n,$

\begin{equation*}
\end{equation*}

\begin{equation*} a(v_{i}, v_{j}) = \begin{cases}
\dfrac{1}{\sqrt{(\alpha_{i}+r_{i})(\alpha_{j}+r_{j})}}J_{n_{i}\times n_{j}} ,& \text{if}~  v_{i}\sim v_{j}~  \text{in}~ G\\
\textbf{0}_{n_{i}\times n_{j}}, & \text{otherwise.}
\end{cases}
\end{equation*}

$ A(G_i)$ is the adjacency matrix of $ G_{i}$, $J_{n_i\times n_{j}}$ is the matrix having all entries $ 1 $, $ \textbf{0}_{n_{i}\times n_{j}} $ is the zero matrix of order $ n_{i}\times n_{j} $ and $ I_{n_i}$ is the identity matrix of order $n_i$.

As each $ G_{i} $ is an $ r_{i} $ regular graph, so the all one vector $e_{n_i}=(\underbrace{1,1,\dots,1}_{n_{i}})^T$  is the eigenvector of the adjacency matrix $ A(G_{i}) $ corresponding to the eigenvalue $r_{i} $ and all other eigenvectors are orthogonal to $e_{n_i}.$ Let $ \lambda_{ik}$, $2\leq k\leq n_i$, be any eigenvalue of $ A(G_{i})$ with the corresponding eigenvector $X=(x_{i1},x_{i2},\dots,x_{in_i})^T$ satisfying $e_{n_i}^TX=0.$ Clearly, the column vector $X$ can be regarded as a function defined on $ V(G_i) $ assigning the vertex $ v_{ij} $ to $ x_{ij} $, that is, $ X(v_{ij})=x_{ij} $ for $ i=1,2,\ldots,n $ and $j=1,2,\dots,n_i$. Now, consider the vector $Y=(y_{1},y_{2},\dots,y_{n})^T$, where

\begin{equation*}
\begin{array}{rl}
x_{ij}&~\text{if}~ v_{ij}\in V(G_i)\\
\end{array}\right.
\end{equation*}

Since, $ e_{n_{i}}^{T}X=0 $ and coordinates of the  vector $Y$ corresponding to vertices in $\cup_{j\ne i}V_j$ of $H$ are zeros,  we have

\begin{align*}
\mathcal{L}(H)Y&=\begin{pmatrix}
\vdots\\
X-\dfrac{1}{r_{i}+\alpha_{i}}\lambda_{ik} X\\
\vdots\\
\end{pmatrix}\\
&=\left (1-\dfrac{1}{r_{i}+\alpha_{i}}\lambda_{ik}\right )Y.
\end{align*}

This shows that $Y$ is an eigenvector of $\mathcal{L}(H) $ corresponding to the eigenvalue

\[ 1-\dfrac{1}{r_{i}+\alpha_{i}}\lambda_{ik}, \]
for every eigenvalue $\lambda_{ik}$, $2\leq k\leq n_i$, of $A(G_i)$. In this way, we have obtained
\[  \sum\limits_{i=1}^{n}n_{i}-n=N-n  \]
eigenvalues. The remaining $ n $ normalized Laplacian eigenvalues of $ H $ are the eigenvalues of the equitable quotient matrix

\begin{equation}\label{Qmat of joined union}
\begin{pmatrix}
\dfrac{\alpha_{1}}{\alpha_1+r_1}& \dfrac{-n_2a_{12}}{\sqrt{(r_{1}+\alpha_{1})(r_{2}+\alpha_{2})}}&\ldots & \dfrac{-n_na_{1n}}{\sqrt{(r_{1}+\alpha_{1})(r_{n}+\alpha_{n})}}\\
\dfrac{-n_1a_{21}}{\sqrt{(r_{2}+\alpha_{2})(r_{1}+\alpha_{1})}}&\dfrac{\alpha_{2}}{\alpha_2+r_2}& \ldots & \dfrac{-n_na_{2n}}{\sqrt{(r_{2}+\alpha_{2})(r_{n}+\alpha_{n})}}\\
\vdots &\vdots  &\ddots &\vdots\\
\dfrac{-n_1a_{n1}}{\sqrt{(r_{n}+\alpha_{n})(r_{1}+\alpha_{1})}}& \dfrac{-n_2a_{n2}}{\sqrt{(r_{n}+\alpha_{n})(r_{2}+\alpha_{2})}}&\ldots &\dfrac{\alpha_{n}}{\alpha_n+r_n}
\end{pmatrix},
\end{equation}

where, for $i\ne j$,

\begin{equation*}
1, & v_i\sim v_j\\
0, & \text{otherwise}.

\end{cases}
\end{equation*}

The next observation is a consequence of Theorem <ref> and gives the normalized Laplacian eigenvalues of the complete multipartite graph.

The normalized Laplacian eigenvalues of the complete $p$-partite graph $K_{n_1,n_2,\dots,n_p}=K_{p}[\overline{K}_{n_1},\overline{K}_{n_2},\dots,\overline{K}_{n_p}]$ with $N=\displaystyle\sum_{i=1}^{p}n_{i}$ consists of the eigenvalue $1$ with multiplicity $N-p$ and the remaining $ p $ eigenvalues are given by the matrix

\begin{equation*}
\begin{pmatrix}
1&\dfrac{-n_{2}}{\sqrt{\alpha_{1}\alpha_{2}}}&\ldots & \dfrac{-n_{p}}{\sqrt{\alpha_{1}\alpha_{p}}}\\
\dfrac{-n_{1}}{\sqrt{\alpha_{2}\alpha_{1}}}&1& \ldots & \dfrac{-n_{p}}{\sqrt{\alpha_{2}\alpha_{p}}}\\
\vdots &\vdots  &\ddots &\vdots\\
\dfrac{-n_{1}}{\sqrt{\alpha_{p}\alpha_{2}}}& \dfrac{-n_{2}}{\sqrt{\alpha_{p}\alpha_{2}}}&\ldots &1
\end{pmatrix}.
\end{equation*}

Proof. This follows from Theorem <ref>, by taking $ G_{i}=\overline{K}_{i} $ and $ \lambda_{ik}(G_{i})=0 $ for each $ i $ and each $ k $.

In particular, if partite sets are of equal size, say $ n_{1}=n_{2}=\dots= n_{p}=t $, then we have the following observation.

Let $ G= K_{t,t,\dots,t} $ be a complete $ p$-partite graph of order $ N=pt .$ Then the normalized Laplacian eigenvalues of $ G $ consists of the eigenvalue $1 $ with multiplicity $ N-p $, the eigenvalue $\dfrac{p}{p-1}$ with multiplicity $p-1$ and the eigenvalue $ 0 $.

Proof. By Theorem <ref>, we have $ \alpha_{i}=t(p-1) $, for $ i=1,2,\dots,p $. Also, by Corollary <ref>, we see that $ 1 $ is an eigenvalue with multiplicity $ pt-p $ and other eigenvalues are given by

\begin{equation*}
1&\dfrac{-1}{p-1}&\ldots & \dfrac{-1}{p-1}\\
\dfrac{-1}{p-1}&1& \ldots & \dfrac{-1}{p-1}\\
\vdots &\vdots  &\ddots &\vdots\\
\dfrac{-1}{p-1}& \dfrac{-1}{p-1}&\ldots &1
\end{pmatrix}.
\end{equation*}

By simple calculations, we see that the normalized Laplacian eigenvalues of matrix $ M_{p} $ consists of the eigenvalue $ \dfrac{p}{p-1} $ with multiplicity $ p-1 $ and simple eigenvalue $ 0. $

Another consequence of Theorem  <ref>, gives the normalized Laplacian eigenvalues of the join of two regular graphs.

Let $ G_{i}$ be an $ r_{i} $ regular graph of order $ n_{i} $ for $i=1,2$. Let $ \lambda_{ik}, 2\leq k\leq n_{i}, i=1,2 $ be the adjacency eigenvalues of $ G_{i} $. Then the normalized Laplacian eigenvalues of $ G=G_{1}\triangledown G_{2} $ consists of the eigenvalue $ 1-\dfrac{1}{r_{1}+n_{2}}\lambda_{1k}A(G_{1}) $, $k=2,\dots, n_{1} $, the eigenvalues $  1-\dfrac{1}{r_{2}+n_{1}}\lambda_{2k}A(G_{1}), k=2,\dots, n_{2} $ and the remaining two eigenvalues are given by the quotient matrix

\begin{equation} \label{Qmat of join of two graphs}
\begin{pmatrix}
\dfrac{n_{2}}{r_{1}+n_{2}}& \dfrac{-n_{2}}{\sqrt{(r_{1}+n_{2})(r_{2}+n_{1})}}\\
\dfrac{-n_{1}}{\sqrt{(r_{1}+n_{2})(r_{2}+n_{1})}} & \dfrac{n_{1}}{r_{2}+n_{1}}
\end{pmatrix}.
\end{equation}

Since $ G_{1} $ and $ G_{2} $ are regular graphs, we observe that the two eigenvalues of matrix (<ref>) are the largest and the smallest normalized Laplacian eigenvalue of $ G=G_{1}\triangledown G_{2} $.

The largest and the smallest normalized Laplacian eigenvalues of $ G_{1}\triangledown G_{2} $ are the eigenvalues of the matrix (<ref>).

(i)  The normalized Laplacian eigenvalues of the complete bipartite graph $ K_{a,b}=K_{a}\triangledown K_{b} $ are

\[
\left  \{ 0,1^{[a+b-2]},2 \right \}.
\]

(ii)  The normalized Laplacian eigenvalues of the complete split graph $CS_{\omega,n-\omega}=K_{\omega}\triangledown \overline{K}_{n-\omega}$, with clique number $ \omega $ and independence number $ n-\omega $ are

\[
\left  \{ 0,\left (\dfrac{n}{n-1}\right)^{[\omega-1]},\dfrac{2n-\omega+1}{n-1} \right \}.
\]

(iii) The normalized Laplacian eigenvalues of the cone graph $ C_{a,b}=C_{a}\triangledown \overline{K}_{b} $ consists of the eigenvalues $ 1-\dfrac{1}{2+b}2\cos\left( \dfrac{2\pi k}{n}\right) $, where $ k=2,\dots,n-1 $, the simple eigenvalues $ 0 $ and $ \dfrac{2b+2}{b+2} $.

(iv)  The normalized Laplacian eigenvalues of the wheel graph $ W_{n}=C_{n-1}\triangledown K_{1} $ consists of the eigenvalues $ 1-\dfrac{2}{3}\cos\left( \dfrac{2\pi k}{m}\right) $, where $ k=2,\dots,n-2 $, and the eigenvalues $ \left \{0,\dfrac{4}{3}\right \}. $

Proof. $ \textbf{(i)} $. This follows from Corollary <ref>,  by taking $ n_{1}=a,n_{2}=b, r_{1}=r_{2}=0 $ and $ \lambda_{1k}=0,$ for $ k=2,\dots,a$ and $ \lambda_{2k} =0$ for each $ k=2,\dots,b$.

$ \textbf{(ii)} $. We recall that the adjacency spectrum of $ K_{\omega} $ is $ \{ \omega-1,-1^{(\omega-1)} \} $. Now, the result follows from Corollary <ref> by taking $ n_{1}=\omega, n_{2}=n-\omega, r_{1}=\omega-1, r_{2}=0, \lambda_{1k}=-1,$ for $ k=2,\dots,\omega $ and $ \lambda_{2k}=0$ for $k=2,3,\dots,n-\omega .$

$ \textbf{(iii)} $. Since adjacency spectrum of $ C_{n} $ is $ \left\lbrace 2\cos\left( \frac{2\pi k}{n}\right) : k=1,2,\dots,n\right\rbrace  $, by taking $ n_{1}=a,n_{2}=b, r_{1}=2, r_{2}=0 $ and $ \lambda_{1k} =2\cos\left( \frac{2\pi i}{m}\right )$ for $ k=2,3,\dots,a-1 $ and $ \lambda_{2,k} $ for $ k=2,\dots,b-1 $ in Corollary <ref>, we get the required eigenvalues.

$ \textbf{(iv)} $. This is a special case of part (iii) with $ a=n-1 $ and $ b=1 .$

A friendship graph $ F_{n} $ is a graph of order $ 2n+1 $, obtained by joining $ K_{1} $ with $ n $ copies of $ K_{2} $, that is, $ F_{n}=K_{1}\triangledown (nK_{2}) =K_{1,n}[K_{1},\underbrace{K_{2},K_{2},\dots,K_{2}}_{n}]$, where $K_1$ corresponds to the root vertex (vertex of degree greater than one) in $K_{1,n}$.  In particular, replacing some of $ K_{2} $'s by $ K_{1} $'s in $F_n$ we get firefly type graph denoted by $ F_{p,n-p}$ and written as

\begin{equation*}
\end{equation*}

A generalized or multi-step wheel network $ W_{a,b} $ is a graph derived from $ a $ copies of $ C_{b} $ and $ K_{1} $, in a such a way that all vertices of each $ C_{b} $ are adjacent to $ K_{1} $. Its order is $ ab+1 $ and can be written as $ W_{a,b}=K_{1}\triangledown (aC_{b})= K_{1,a}[K_{1},\underbrace{C_{b},\dots,C_{b}}_{a} ]$.

The normalized Laplacian eigenvalues of the friendship graph $ F_{n} $, the firefly type graph $ F_{p,n-p}$ and $ W_{a,b} $ are given by the following result.

(i) The normalized Laplacian eigenvalues of $ F_{n} $ are

\[ \bigg\{0, \left ( \dfrac{1}{2} \right )^{n-1}, \left ( \dfrac{3}{2} \right )^{n+1} \bigg\}. \]

(ii) The normalized Laplacian eigenvalues of $ F_{p,n-p} $  are

\[
\bigg\{0, \left ( \dfrac{1}{2} \right )^{[n-p-1]}, 1^{[p-1]}, \left ( \dfrac{3}{2} \right )^{[n-p]}, \dfrac{5\sqrt{2n-p}\pm\sqrt{2n+7p}}{4\sqrt{2n-p}} \bigg\}.
\]

(iii) The normalized Laplacian eigenvalues of $ W_{a,b} $ consists of the eigenvalues $ 0$, the eigenvalue $ \dfrac{4}{3} $ and the eigenvalues $ 1-\dfrac{2}{3}\cos\left (\dfrac{2\pi k}{b}\right ),$ for  $ k=2,\dots,b. $

Proof. (i). By Theorem <ref> and the definition of $ F_{n} $, we have

\[ \alpha_{1}=2n, \alpha_{2}=\dots=\alpha_{n+1}=1~~ \text{and}~~ r_{1}=0, r_{2}=\dots =r_{n+1}=1. \]

So, by Theorem <ref>, we see that $ \dfrac{3}{2} $ is the normalized Laplacian eigenvalues of $ F_{n} $ with multiplicity $ n. $ The remaining eigenvalues are given by the block matrix

\begin{equation}\label{Qmat of Fn}
\left (\begin{array}{c|c c c c}
1 & \dfrac{-1}{\sqrt{n}}& \dots & \dfrac{-1}{\sqrt{n}}&\dfrac{-1}{\sqrt{n}}\\
\hline
\dfrac{-1}{2\sqrt{n}} & \dfrac{1}{2} & \dots & 0& 0\\
\vdots & \vdots &\ddots& \vdots & \vdots\\
\dfrac{-1}{2\sqrt{n}}& 0 & \dots &  \dfrac{1}{2} & 0\\
\dfrac{-1}{2\sqrt{n}}& 0 & \dots & 0 & \dfrac{1}{2}
\end{array}\right ).
\end{equation}

Clearly, $ \dfrac{1}{2} $ is the normalized laplacian eigenvalue of (<ref>) with multiplicity $ n-1 $ and the remaining two eigenvalues of block matrix (<ref>) are given by the quotient matrix

\[ \begin{pmatrix}
1 & \frac{-n}{\sqrt{n}}\\
\dfrac{-1}{2\sqrt{n}} & \dfrac{1}{2}

\end{pmatrix}.
\]

First we will prove (iii) and then (ii).

(iii). As in part (iii) of Proposition <ref>, we see that $ 1-\dfrac{2}{3}\cos\left (\dfrac{2\pi k}{b}\right ),$ for  $ k=2,\dots,b. $ are the normalized Laplacian eigenvalues of $ W_{a,b} $. The other eigenvalues are given by the block matrix

\begin{equation*}
\left( \begin{array}{c| c c c c}
1 & \dfrac{-b}{\sqrt{3ab}}& \dots & \dfrac{-b}{\sqrt{3ab}}&\dfrac{-b}{\sqrt{3ab}}\\
\hline
\dfrac{-1}{\sqrt{3ab}} & \dfrac{1}{3} & \dots & 0 &  0\\
\vdots & \vdots &\ddots& \vdots & \vdots\\
\dfrac{-1}{\sqrt{3ab}} & 0 & \dots& \dfrac{1}{3} & 0\\
\dfrac{-1}{\sqrt{3ab}}& 0 & \dots & 0 & \dfrac{1}{3}
\end{array}\right).
\end{equation*}

Now, as in part (i), $ \bigg\{0,\left (\dfrac{1}{3}\right )^{a-1}, \dfrac{4}{3}\bigg\} $ are the remaining normalized Laplacian eigenvalues of $ W_{a,b} $.

(ii). Since $ \alpha_{1}=p+2(n-p)=2n-p $ and $ \alpha_{2}=\dots=\alpha_{2n+1-p}=1 $, so by Theorem <ref>, with $ r_{1}=\dots=r_{p+1}=0, ~ r_{p+2}=\dots=r_{2n+1-p}=1,$ we see that $ \dfrac{3}{2} $ is the normalized Laplacian eigenvalue of $ F_{p,n-p} $ with multiplicity $ n-p $. The other normalized Laplacian eigenvalues of $ F_{p,n-p} $ are given by the block matrix

\begin{equation}\label{Qmat Fp,n-p}
\left(
\begin{array}{c| c c c| c c c}
1 & \dfrac{-1}{\sqrt{2n-p}} & \dots & \dfrac{-1}{\sqrt{2n-p}}& \dfrac{-2}{\sqrt{2(2n-p)}}& \dots & \dfrac{-2}{\sqrt{2(2n-p)}}\\
\hline
\dfrac{-1}{\sqrt{2n-p}} & 1 & \dots & 0 & 0 &\dots & 0\\
\vdots & \vdots & \ddots & \vdots & \vdots &  \dots & \vdots\\
\dfrac{-1}{\sqrt{2n-p}} & 0 & \dots & 1 & 0 & \dots & 0\\

\hline
\dfrac{-2}{\sqrt{2(2n-p)}} & 0 & \dots & 0 & \dfrac{1}{2} & \dots & 0\\
\vdots & \vdots & \dots & \vdots & \vdots &  \ddots & \vdots\\
\dfrac{-2}{\sqrt{2(2n-p)}} & 0 & \dots & 0 & 0 & \dots & \dfrac{1}{2}\\
\end{array}
\right).
\end{equation}

By simple calculations, $ 1 $ and $ \dfrac{1}{2} $ are the normalized Laplacian eigenvalues of (<ref>) and the remaining eigenvalues of block matrix (<ref>) are given by the quotient matrix

\begin{equation}\label{qqmat}
\begin{pmatrix}
1 & \dfrac{-p}{\sqrt{2n-p}} & \dfrac{-2(n-p)}{\sqrt{2(2n-p)}}\\
\dfrac{-1}{\sqrt{2n-p}} & 1 & 0\\
\dfrac{-1}{\sqrt{2(2n-p)}} & 0 & \dfrac{1}{2}
\end{pmatrix}.
\end{equation}

Now, it is easy to see that $ 0 $ and $ \dfrac{5\sqrt{2n-p}\pm\sqrt{2n+7p}}{4\sqrt{2n-p}} $ are the normalized Laplacian eigenvalues of quotient matrix (<ref>).

§ NORMALIZED LAPLACIAN EIGENVALUES OF THE POWER GRAPHS OF CYCLIC GROUP $ \MATHBB{Z}_{N} $

In this section, we consider the power graphs of finite cyclic group $ \mathbb{Z}_{n} $. As an application to Theorem <ref> and its consequences obtained in Section <ref>, we determine the normalized Laplacian eigenvalues of power graph of $ \mathbb{Z}_{n}. $

All groups are assumed to be finite and every cyclic group of order $ n $ is taken as isomorphic copy of integral additive modulo group $ \mathbb{Z}_{n} $ with identity denoted by $ 0 $. Let $ \mathcal{G} $ be a finite group of order $ n $ with identity  $ e $. The power graph of group $ \mathcal{G} $, denoted by $ \mathcal{P}(\mathcal{G}) $, is the simple graph with vertex set as the elements of group $ \mathcal{G} $ and two distinct vertices $ x,y\in \mathcal{G} $ are adjacent if and only if one is the positive power of the other, that is, $ x\sim y $ if and only if $ x^{i}=y $ or $ y^{j}=x $, for positive integers $i,j$ with  $ 2\leq i,j\leq n $. These graphs were introduced in [100], see also [23]. Such graphs have valuable applications and are related to automata theory [99], besides being useful in characterizing finite groups.  We let $ U_{n}^{*}=\{x\in\mathbb{Z}_{n}: (x,n)=1 \}\cup\{0\} $, where $ (x,n) $ denotes greatest common divisor of $ x $ and $ n $. Our other group theory notations are standard and can be taken from [115]. More work on power graphs can be seen in [19, 23, 111, 1, 27] and the references therein.

The adjacency spectrum, the Laplacian and the signless Laplacian spectrum of power graphs of finite cyclic and dihedral groups have been investigated in  [12, 25, 89, 112, 119]. The normalized Laplacian eigenvalues of power graphs of certain finite groups were studied in [88].

Let $ n $ be a positive integer. Then $ \tau(n) $ denotes the number of positive factors of $ n $, that is
\[ \tau(n)=\sum\limits_{d|n} 1, \]
where $ d|n $ denotes $ d $ divides $ n $.

The Euler's totient function $ \phi(n) $ denotes the number of positive integers less or equal to $ n $ and relatively prime to $ n $.

We say $ n $ is in canonical decomposition if $n=p_{1}^{n_{1}}p_{2}^{n_{2}}\dots p_{r}^{n_{r}} $, where $ r,n_{1},n_{2},\dots,n_{r} $ are positive integers and $ p_{1},p_{2},\dots,p_{r} $ are distinct primes.

The following result counts the values of $ \tau(n) $.

Let $ n $ be a positive integer with canonical decomposition $ n=p_{1}^{n_{1}}p_{2}^{n_{2}}\dots p_{r}^{n_{r}} $. Then

\[ \tau(n)= (n_{1}+1) (n_{2}+1)\dots(n_{r}+1) \]

The following result gives some properties of Euler's totient function.

Let $ \phi $ be the Euler's totient function. Then following hold.

(i) $ \phi $ is multiplicative, that is
$ \phi(st)=\phi(s)\phi(t), $ whenever $ s $ and $ t $ are relatively prime.

(ii) Let $ n $ be a positive integer. Then
$ \sum\limits_{d|n}\phi(d)=n. $

(iii) Let $ p $ be a prime. Then
$ \sum\limits_{i=1}^{l}\phi(p^{l})=p^{l}-1. $

Let $ n $ be a positive integer and $ d $ divides $ n $. The divisor $ d $ is the proper divisor of $ n $, if $ 1<d<n . $ Let $\mathbb{G}_{n}$ be a simple graph with vertex set as the proper divisor set $ \{d_{i}: 1,n\neq d_{i}|n,  ~ 1 \leq i \leq t\} $ and edge set $\{ d_{i}d_{j}: d_{i}|d_{j}, ~ 1 \leq i< j\leq t\} $, for $ 1\leq i<j\leq t $. If the canonical decomposition of $n$ is $ n=p_{1}^{n_{1}}p_{2}^{n_{2}}\dots p_{r}^{n_{r}} $, then by Theorem <ref>, the order of the graph  $\mathbb{G}_{n}$ is  $ |V( \mathbb{G}_{n})|=\prod\limits_{i=1}^{r}(n_{i}+1)-2. $ Also, $\mathbb{G}_{n}$ is a connected graph [134], provided $ n $ is neither a prime power nor the product of two distinct primes. In [111], $\mathbb{G}_{n}$ is used as the underlying graph for studying the power graph of finite cyclic group $ \mathbb{Z}_{n} $ and it has been shown that for each proper divisor $ d_{i} $ of $ n $, $ \mathcal{P}(\mathbb{Z}_{n})$ has a complete subgraph of order $ \phi(d_{i}) $.

The following theorem shows that $ \dfrac{n}{n-1} $ is always the normalized Laplacian eigenvalue of the power graph $ \mathcal{P}(\mathbb{Z}_{n})$.

Let $ \mathbb{Z}_{n} $ be a finite cyclic group of order $ n\geq 3 $. Then $ \dfrac{n}{n-1} $ is normalized Laplacian eigenvalue of $ \mathcal{P}(\mathbb{Z}_{n}) $ with multiplicity at least $ \phi(n)$.

Proof. Let $ \mathbb{Z}_{n} $ be the cyclic group of order $ n\geq 3 $. Then the identity $ 0 $ and invertible elements of  the group $ \mathbb{Z}_{n} $ in the power graph $ \mathcal{P}(\mathbb{Z}_{n}) $ are adjacent to every other vertex in $ \mathcal{P}(\mathbb{Z}_{n}) $. Since it is well known that the number of invertible elements of $ \mathbb{Z}_{n} $ are $ \phi(n) $ in number, so the induced power graph $ \mathcal{P}(U_{n}^{*}) $ is the complete graph $ K_{\phi(n)+1} $. Thus, by Theorem <ref>, we see that $ \mathcal{P}(\mathbb{Z}_{n})=K_{\phi(n)+1}\triangledown \mathcal{P}(\mathbb{Z}_{n}\setminus U_{n}^{*})$. By applying Corollary <ref>, we get
\begin{equation*}
\end{equation*}

as the normalized Laplacian eigenvalue with multiplicity at least $ \phi(n) $, since $ \dfrac{n}{n-1} $ can also be the normalized Laplacian eigenvalue of quotient matrix (<ref>).

If $ n=p^{z} $, where $ p $ is prime and $ z $ is a positive integer, then we have following observation.

If $ n=p^{z} $, where $ p $ is prime and $ z $ is a positive integer, then the normalized Laplacian eigenvalues of  $\mathcal{P}(\mathbb{Z}_n) $ are

\[  \left \{ 0, \left (\dfrac{n}{n-1}\right )^{[(n-1)]} \right \}. \]

Proof. If $ n=p^z $, where $ p $ is prime and $ z $ is a positive integer, then as shown in [23], $ \mathcal{P}(\mathbb{Z}_n)$ is isomorphic to the complete graph $ K_{n}$ and hence the result follows.

The next observation gives the normalized Laplacian eigenvalues of $ \mathcal{P}(\mathbb{Z}_n) $, when $ n $ is product of two primes.

Let $ n=pq$ be product of two distinct primes. Then the normalized Laplacian eigenvalues of $ \mathcal{P}(\mathbb{Z}_n) $ are

\[ \left \{ 0,\left (\dfrac{n}{n-1}\right )^{[\phi(n)]}, \left (1+\dfrac{1}{q\phi(p)}\right )^{[\phi(p)-1]}, \left (1+\dfrac{1}{p\phi(q)}\right )^{[\phi(q)-1]} \right  \} \]
and the zeros of polynomial

\begin{align*}
p(x)=x\Bigg(x^2&-x\left (\dfrac{\phi(n)+1}{q\phi(p)}+\dfrac{\phi(p)+\phi(q)}{q\phi(p)+\phi(q)}+\dfrac{\phi(n)+1}{p\phi(q)}\right )+\dfrac{(\phi(n)+1)\phi(p)}{p\phi(q)(q\phi(p)+\phi(q))} \\
&+\dfrac{(\phi(n)+1)^{2}}{n\phi(n)}+\dfrac{(\phi(n)+1)\phi(q)}{q\phi(p)(q\phi(p)+\phi(q))} \Bigg).
\end{align*}

Proof. If $ n=pq $, where $p$ and $q$, ($ p<q) $) are primes, then $ \mathcal{P}(\mathbb{Z}_{n}) $ [27] can be written as

\[  \mathcal{P}(\mathbb{Z}_n)=(K_{\phi(p)}\cup K_{\phi(q)})\triangledown K_{\phi(n)+1}= P_{3}[K_{\phi(p)},K_{\phi(pq)+1},K_{\phi(q)} ] . \]

By Theorem <ref>, $ \dfrac{n}{n-1} $ is the normalized Laplacian eigenvalue with multiplicity $ \phi(n) $. Again, by Theorems <ref> and <ref>, we see that $ \dfrac{1}{q\phi(p)} $ and $ \dfrac{1}{q\phi(p)} $ are the normalized Laplacian eigenvalues of $ \mathcal{P}(\mathbb{Z}_n) $ with multiplicity $ \phi(p)-1 $ and $ \phi(q)-1 $, respectively. The remaining three normalized Laplacian eigenvalues are given by the following matrix

\begin{equation*} \label{Qmat pq}
\begin{pmatrix}
\dfrac{\phi(n)+1}{q\phi(p)} & \dfrac{-(\phi(n)+1)}{\sqrt{q\phi(p)(q\phi(p)+\phi(q))}} & 0\\
\dfrac{-\phi(p)}{\sqrt{q\phi(p)(q\phi(p)+\phi(q))}} &\dfrac{\phi(p)+\phi(q)}{q\phi(p)+\phi(q)} & \dfrac{-\phi(q)}{\sqrt{p\phi(q)(q\phi(p)+\phi(q))}}\\
0 & \dfrac{-(\phi(n)+1)}{\sqrt{p\phi(q)(q\phi(p)+\phi(q))}} & \dfrac{\phi(n)+1}{p\phi(q)}
\end{pmatrix}.
\end{equation*}

By Corollaries <ref> and <ref>, we have the following proposition.

Equality holds in Theorem (<ref>), if n is some prime or product of two primes.

The following theorem [111] shows that the power graph of a finite cyclic group $ \mathbb{Z}_{n} $ can be written as the joined union, each of whose components are cliques.

If $ \mathbb{Z}_{n} $ is a finite cyclic group of order $n\ge 3$, then the power graph $\mathcal{P}(\mathbb{Z}_{n})$ is given by

\begin{equation*}
\mathcal{P}(\mathbb{Z}_{n}) =K_{\phi(n)+1}\triangledown \mathbb{G}_{n}[K_{\phi(d_{1})},K_{\phi(d_{2})},\dots,K_{\phi(d_{t})}],
\end{equation*}

where $\mathbb{G}_{n}$ is the graph of order $t$ defined above.

Using Theorem <ref> and its consequences, we can compute the normalized Laplacian eigenvalues of $ \mathcal{P}(\mathbb{Z}_{n}) $ in terms of the adjacency spectrum of $ K_{\omega} $ and zeros of the characteristic polynomial of the quotient matrix.

We form a connected graph $ H=K_{1}\triangledown \mathbb{G}_{n} $ which is of diameter at most two if $ \mathbb{G}_{n} $ is not complete, otherwise its diameter is $ 1 $. In the following result, we compute the normalized Laplacian eigenvalues of the power graph of $ \mathbb{Z}_{n} $ by using Theorems <ref> and <ref>.

The normalized Laplacian eigenvalues of $ \mathcal{P}(\mathbb{Z}_n) $ are

\begin{equation*}
\left\lbrace \left (\dfrac{n}{n-1}\right )^{(\phi(n))}, \left (\dfrac{\phi(d_{1})+\alpha_{2}}{\phi(d_{1})+\alpha_{2}-1}\right )^{[\phi(d_{1})-1]},\dots, \left (\dfrac{\phi(d_{t})+\alpha_{r+1}}{\phi(d_{t})+\alpha_{t+1}-1}\right )^{[\phi(d_{t})-1]} \right\rbrace
\end{equation*}

and the $ t+1 $ eigenvalues of  matrix (<ref>)

\begin{equation}\label{quotient matrix of Z_n}
\begin{pmatrix}
\dfrac{n-1-\phi(n)}{n-1}& \dfrac{-\phi(d_{1})a_{12}}{\sqrt{(\phi(n)+\alpha_{1})(r_{2}+\alpha_{2})}}&\ldots & \dfrac{-\phi(d_{t})a_{1(t+1)}}{\sqrt{b}}\\
\dfrac{-\phi(d_{1})a_{21}}{\sqrt{(r_{2}+\alpha_{2})(\phi(n)+\alpha_{1})}}&\dfrac{\alpha_{2}}{\alpha_2+r_2}& \ldots & \dfrac{-\phi(d_{t})a_{2(t+1)}}{\sqrt{(r_{2}+\alpha_{2})(r_{t+1}+\alpha_{t+1})}}\\
\vdots &\vdots  &\ddots &\vdots\\
\dfrac{-\phi(d_{1})a_{(t+1)1}}{\sqrt{(r_{n}+\alpha_{t+1})(\phi(n)+\alpha_{1})}}& \dfrac{-\phi(d_{2})a_{(t+1)2}}{\sqrt{(r_{n}+\alpha_{t+1})(r_{2}+\alpha_{2})}}&\ldots &\dfrac{\alpha_{t+1}}{\alpha_{t+1}+r_{t+1}}
\end{pmatrix},
\end{equation}

where, for $i\ne j$,

\begin{equation*}
1, & v_i\sim v_j\\
0, & v_i\nsim v_j

\end{cases},
\end{equation*}

$ b=(\phi(n)+\alpha_{1})(r_{t+1}+\alpha_{t+1}) $
and $ r_{i}=\phi(d_{i})-1, ~ \text{for} ~i=2,\dots, t+1. $

Proof. Let $ \mathbb{Z}_{n} $ be a finite cyclic group of order $ n $. Since the identity element $ 0 $ and the $ \phi(n) $ generators of the group $ \mathbb{Z}_{n} $ are adjacent to every other vertex of $ \mathcal{P}(\mathbb{Z}_{n}) $, therefore, by Theorem <ref>, we have

\begin{equation*}
\mathcal{P}(\mathbb{Z}_{n}) =K_{\phi(n)+1}\triangledown \mathbb{G}_{n}[K_{\phi(d_{1})},K_{\phi(d_{2})},\dots,K_{\phi(d_{t})}]= H[K_{\phi(n)+1},K_{\phi(d_{1})},K_{\phi(d_{2})},\dots,K_{\phi(d_{t})}],
\end{equation*}

where $ H=K_{1}\triangledown \mathbb{G}_{n} $ is the graph with vertex set $ \{v_{1},\dots, v_{t+1}\} $. Taking $ G_{1}=K_{\phi(n)+1}$ and $ G_{i}=K_{\phi(d_{i-1})} $, for $ i=2,\dots,t+1 $, in  Theorem <ref> and using the fact that the adjacency spectrum of $K_{\omega}$ consists of the eigenvalue $\omega-1$ with multiplicity $1$ and the eigenvalue $-1$ with multiplicity $\omega-1$, it follows that

\begin{equation*}
1-\dfrac{1}{r_{1}+\alpha_{1}}\lambda_{1k}A(G_{1}) =1-\dfrac{1}{r_{1}+\alpha_{1}}(-1)= 1+\dfrac{1}{\phi(n)+n-\phi(n)-1}=\dfrac{n}{n-1}
\end{equation*}

is a normalized eigenvalue of $ \mathcal{P}(\mathbb{Z}_n) $ with multiplicity $ \phi(n)$. Note that we have used the fact that vertex $ v_{1} $ of graph $ H $ is adjacent to every other vertex of $ H $ and $ \alpha_1=\sum\limits_{ d|n, d\ne 1,n}\phi(d)=n-1-\phi(n)$, as $ \sum\limits_{ d|s}\phi(d) =s.$ Similarly, we can show that $ \dfrac{\phi(d_{1})+\alpha_{2}}{\phi(d_{1})+\alpha_{2}-1},\dots, \dfrac{\phi(d_{t})+\alpha_{t+1}}{\phi(d_{t})+\alpha_{t+1}-1} $ are the normalized Laplacian eigenvalues of $ \mathcal{P}(\mathbb{Z}_n) $ with multiplicities $ \phi(d_{1})-1, \dots, \phi(d_{t})-1 $, respectively.  The remaining normalized Laplacian eigenvalues are the eigenvalues of quotient matrix  (<ref>).

From Theorem <ref>, it is clear that all the normalized Laplacian eigenvalues of the power graph $ \mathcal{P}(\mathbb{Z}_{n})$ are completely determined except the $t+1$ eigenvalues, which are the eigenvalues of matrix (<ref>). Further, it is also clear that matrix (<ref>)  depends upon the structure of the graph $\mathbb{G}_{n} $, which is not known in general. However, if we give some particular value to $n$, then it may be possible to know the structure of graph $\mathbb{G}_{n}$ and hence about matrix (<ref>). This information may be helpful to determine the $t+1$ remaining normalized Laplacian eigenvalues of the power graph $ \mathcal{P}(\mathbb{Z}_{n})$.

We discuss some particular cases of Theorem <ref>.

[fill=black] (0,0) circle (2pt);[fill=black] (2,0) circle (2pt);[fill=black] (4,0) circle (2pt);
[fill=black] (-1,-1) circle (2pt);[fill=black] (1,-1) circle (2pt);[fill=black] (3,-1) circle (2pt);

[thin] (0,0)–(-1,-1); [thin] (0,0)–(1,-1);[thin] (2,0)–(-1,-1);
[thin] (2,0)–(3,-1); [thin] (4,0)–(1,-1);[thin] (4,0)–(3,-1);

at (0,.3) $ p $;at (2,.3) $ q $;at (4,.3) $ r $;
at (-1,-1.3) $ pq $;at (1,-1.3) $ pr $;at (3,-1.3) $ qr $;

[fill=black] (0,0) circle (2pt);[fill=black] (2,0) circle (2pt);[fill=black] (4,0) circle (2pt);
[fill=black] (-1,-1) circle (2pt);[fill=black] (1,-1) circle (2pt);[fill=black] (3,-1) circle (2pt);
[fill=black] (2.9,-2.3) circle (2pt);

[thin] (0,0)–(2.9,-2.3);[thin] (2,0)–(2.9,-2.3);[thin] (3,-1)–(2.9,-2.3);
[thin] (2,0)–(2.9,-2.3); [thin] (4,0)–(2.9,-2.3);[thin] (-1,-1)–(2.9,-2.3);

[thin] (0,0)–(-1,-1); [thin] (0,0)–(1,-1);[thin] (2,0)–(-1,-1);
[thin] (2,0)–(3,-1); [thin] (4,0)–(1,-1);[thin] (4,0)–(3,-1);

at (0,.3) $ p $;at (2,.3) $ q $;at (4,.3) $ r $;
at (-1,-1.3) $ pq $;at (1,-1.3) $ pr $;at (3,-0.7) $ qr $; at (2.9,-2.7) $ K_{1} $;

Divisor graph $ \mathbb{G}_{pqr} $ and $ H=K_{1}\triangledown \mathbb{G}_{pqr} $.

Now, let $ n=pqr $, where $p,~ q,~ r$  with $ p<q<r$ are primes. From the definition of $\mathbb{G}_{n}$, the  vertex set and edge set of $\mathbb{G}_{n}$ are $\{ p, q, r, pq, pr, qr \}$ and $\{(p,pq),(p,pr),(q,pq),(q,qr),(r,pr),(r,qr)\} $, respectively, and is shown in Figure $ (4.1) $. Let $ H=K_{1}\triangledown \mathbb{G}_{n}.$ Then

\begin{equation*}
\mathcal{P}(\mathbb{Z}_n)= H[K_{\phi(n)+1}, K_{\phi(p)},K_{\phi(q)},K_{\phi(r)},K_{\phi(pq)},K_{\phi(pr)}, K_{\phi(qr)}].
\end{equation*}

By Theorem <ref>, we have
\begin{align*}
\Big (\alpha_{1},\alpha_{2}, \alpha_{3},\alpha_{4},\alpha_{5},\alpha_{6},\alpha_{7}\Big)=&\Big(n-\phi(n)-1,~\phi(n)+1+\phi(pq)+\phi(pr),~\phi(n)+1\\
&+\phi(r)\Big ).
\end{align*}

Also, for $ i=1,2,\dots,7$,   $ r_{i}^{'}=\alpha_{i}+r_{i} $, we have

\begin{align*}
\Big (r_{1}^{'},r_{2}^{'}, r_{3}^{'}, r_{4}^{'} ,r_{5}^{'},r_{6}^{'},r_{7}^{'}\Big)=& \Big(n-1,~\phi(n)+\phi(p)+\phi(pq)+\phi(pr),~\phi(n)+\phi(q)+\phi(pq)\\
&+\phi(r)\Big ).
\end{align*}

Now, by Theorem <ref>, $ \dfrac{n}{n-1} $ is the normalized Laplacian eigenvalue with multiplicity $ \phi(n) $.  Using the above information and Theorem <ref>, second distinct normalized Laplacian eigenvalue is $ 1+\dfrac{1}{r_{2}+\alpha_{2}}=1+\dfrac{1}{\phi(n)+\phi(p)+\phi(pq)+\phi(pr)} $ with multiplicity $ \phi(p)-1 $. In a similar way, we see that other eigenvalues are

\begin{align*}
\end{align*}

with multiplicities $ \phi(q)-1,~ \phi(r)-1,~ \phi(pq)-1,~ \phi(pr)-1,~ \phi(qr)-1$, respectively. The remaining $7$ eigenvalues are given by the following matrix

\begin{equation*}\label{pqr}
\begin{pmatrix}
z_{1}& -\phi(p)c_{12} & -\phi(q)c_{13} & -\phi(r)c_{14}& -\phi(pq)c_{15}& -\phi(pr)c_{16}& -\phi(qr)c_{17}\\
(\phi(n)+1)c_{21} & z_{2} & 0 & 0& -\phi(pq)c_{25} & -\phi(pr)c_{26}& 0 \\
(\phi(n)+1)c_{31} & 0 & z_{3} & 0& -\phi(pq)c_{35} & 0 & -\phi(qr)c_{37} \\
(\phi(n)+1)c_{41} & 0 & 0 & z_{4} & 0&  -\phi(pr)c_{46} & -\phi(qr)c_{47} \\
(\phi(n)+1)c_{51} & -\phi(p)c_{25} & -\phi(q)c_{35} & 0 & z_{5} & 0  & 0  \\
(\phi(n)+1)c_{61} & -\phi(p)c_{26} & 0 & -\phi(r)c_{64} & 0 & z_{6} & 0  \\
(\phi(n)+1)c_{71} & 0 & -\phi(q)c_{75} & -\phi(r)c_{74} & 0 & 0 &  z_{7}
\end{pmatrix},
\end{equation*}

\begin{align*}
z_{1}=&\frac{n-\phi(n)-1}{n-1}, ~ z_{2}= \frac{\phi(n)+1+\phi(pq)+\phi(pr)}{\phi(n)+\phi(p)+\phi(pq)+\phi(pr)},\\
&~ z_{3}= \frac{\phi(n)+1+\phi(pq)+\phi(qr)}{\phi(n)+\phi(q)+\phi(pq)+\phi(qr)},~z_{4}=\dfrac{\phi(n)+1+\phi(pr)+\phi(qr)}{\phi(n)+\phi(r)+\phi(pr)+\phi(qr)},\\
&z_{5}= \dfrac{\phi(n)+1+\phi(p)+\phi(q)}{\phi(n)+\phi(pq)+\phi(p)+\phi(q)},~ z_{6}= \dfrac{\phi(n)+1+\phi(p)+\phi(r)}{\phi(n)+\phi(pr)+\phi(p)+\phi(r)},\\ &z_{7}=\dfrac{\phi(n)+1+\phi(q)+\phi(r)}{\phi(n)+\phi(qr)+\phi(q)+\phi(r)}\\
\text{and}~c_{ij}&=c_{ji}=\dfrac{1}{\sqrt{(r_{i}+\alpha_{i})(r_{j}+\alpha_{j})}}.
\end{align*}

Next, we discuss the normalized Laplacian eigenvalues of the finite cyclic group $\mathbb{Z}_{n}$, with $n=p^{n_{1}}q^{n_{2}},$ where $ p<q $ are primes and $ n_{1}\leq  n_{2}$ are positive integers. We consider the case when both $ n_{1} $ and $ n_{2} $ are even, and the case when they are odd can be discussed similarly.

Let $ \mathcal{P}(\mathbb{Z}_{p^{n_{1}}q^{n_{2}}}) $ be the power graph of the finite cyclic group $\mathbb{Z}_{p^{n_{1}}q^{n_{2}}}$ of order $ n={p^{n_{1}}q^{n_{2}}} $, where $p<q$ are primes and $n_{1}=2m_{1}\leq n_{2}=2m_{2}$ are even positive integers. Then the normalized Laplacian eigenvalues of $ \mathcal{P}(\mathbb{Z}_{p^{n_{1}}q^{n_{2}}}) $ consists of eigenvalue set

\begin{align*}
&\left (\dfrac{n}{n-1}\right )^{[\phi(n)]}, \left (\dfrac{n-q^{n_{2}}+1}{n-q^{n_{2}}}\right )^{[\phi(p)-1]},\\
&\qquad \vdots\\
&\left (\dfrac{p^{m_{1}-1}+q^{n_{2}}(p^{n_{1}}-p^{m_{1}-1})}{p^{m_{1}-1}+q^{n_{2}}(p^{n_{1}}-p^{m_{1}-1})}-1\right )^{[\phi(p^{m_{1}})-1]},\\
&\left (\dfrac{p^{n_{1}-1}+q^{n_{2}}\phi(p^{n_{1}})}{p^{n_{1}-1}+q^{n_{2}}\phi(p^{n_{1}})-1}\right )^{\left [\phi\left (p^{n_{1}}\right )-1\right ]},\left (\dfrac{n-p^{n_{1}}+1}{n-p^{n_{1}}}\right )^{\left [\phi\left (q\right )-1\right ]},\\
&\left (\dfrac{q^{m_{2}-1}+p^{n_{1}}(q^{n_{2}}-q^{m_{2}-1})}{q^{m_{2}-1}+p^{n_{1}}(q^{n_{2}}-q^{m_{2}-1})-1}\right )^{\left [\phi\left (q^{m_{2}}\right )-1\right ]},\\
&\left (\dfrac{q^{n_{2}-1}+p^{n_{1}}\phi(q^{n_{2}})}{q^{n_{2}-1}+p^{n_{1}}\phi(q^{n_{2}})-1}\right )^{\left [\phi\left (q^{n_{2}}\right )-1\right ]},\left (\dfrac{\phi(p)+\phi(q)+(q^{n_{2}}-1)(p^{n_{1}}-1)+1}{\phi(p)+\phi(q)+(q^{n_{2}}-1)(p^{n_{1}}-1)}\right )^{\left [\phi\left (pq\right )-1\right ]},\\
&\left (\dfrac{q^{n_{2}}(p^{n_{1}}-1)+q^{m_{2}}-q^{m_{2}-1}(p^{n_{1}}-p)}{q^{n_{2}}(p^{n_{1}}-1)+q^{m_{2}}-q^{m_{2}-1}(p^{n_{1}}-p)-1}\right )^{\left [\phi\left (pq^{m_{2}}\right )-1\right ]},\\
& \left (\dfrac{pq^{n_{2}}+\phi(p^{n_{1}})(q^{n_{2}}-q)}{pq^{n_{2}}+\phi(p^{n_{1}})(q^{n_{2}}-q)-1} \right )^{\left [\phi\left (pq^{n_{2}}\right )-1\right ]},\\
&\left (\dfrac{p^{m_{1}}+p^{n_{1}}(q^{n_{2}}-1)-p^{m_{1}-1}(q^{n_{2}}-q)}{p^{m_{1}}+p^{n_{1}}(q^{n_{2}}-1)-p^{m_{1}-1}(q^{n_{2}}-q)-1} \right )^{\left [\phi\left (p^{m_{1}}q\right )-1\right ]},\\
&\left (\dfrac{n+p^{m_{1}}q^{m_{2}}+p^{m_{1}-1}q^{m_{2}-1}-\phi(p^{m_{1}}q^{m_{2}})-p^{n_{1}}q^{m_{2}-1}-p^{m_{1}-1}q^{n_{2}}}{n+p^{m_{1}}q^{m_{2}}+p^{m_{1}-1}q^{m_{2}-1}-\phi(p^{m_{1}}q^{m_{2}})-p^{n_{1}}q^{m_{2}-1}-p^{m_{1}-1}q^{n_{2}}-1} \right )^{\left [\phi\left (p^{m_{1}}q^{m_{2}}\right )-1\right ]},\\
&\left (\dfrac{p^{m_{1}}q^{n_{2}}+\phi(q^{n_{2}})(p^{n_{1}}-p^{m_{1}})}{p^{m_{1}}q^{n_{2}}+\phi(q^{n_{2}})(p^{n_{1}}-p^{m_{1}})-1} \right )^{\left [\phi\left (p^{m_{1}}q^{n_{2}}\right )-1\right ]},\\
&\left (\dfrac{p^{n_{1}}q+\phi(p^{n_{1}})(q^{n_{2}}-q)}{p^{n_{1}}q+\phi(p^{n_{1}})(q^{n_{2}}-q)-1} \right )^{\left [\phi\left (p^{n_{1}}q\right )-1\right ]},\\
& \left ( \dfrac{p^{n_{2}}q^{m_{1}}+\phi(p^{n_{1}})(q^{n_{2}}-q^{m_{1}})}{p^{n_{2}}q^{m_{1}}+\phi(p^{n_{1}})(q^{n_{2}}-q^{m_{1}})-1} \right )^{\left [\phi\left (p^{n_{1}}q^{m_{2}}\right )-1\right ]},\\
&\left ( \dfrac{p^{n_{1}}q^{n_{2}-1}+\phi(n)}{p^{n_{1}}q^{n_{2}-1}+\phi(n)-1} \right )^{\left [\phi\left (p^{n_{1}}q^{n_{2}-1}\right )-1\right ]}\end{align*}

and the remaining eigenvalues are given by matrix (<ref>).

Proof. Suppose that $n=p^{n_{1}}q^{n_{2}}$, where $ n_{1}=2m_{1} $ and $ n_{2}=2m_{2} $ are even with $ n_{1}\leq n_{2} $ and $ m_{1} $ and $ m_{2} $ are positive integers. Since the total number of divisors of $ n $ are $ (n_{1}+1)(n_{2}+1) $, so the order of $ \mathbb{G}_{p^{n_{1}}q^{n_{2}}} $ is $ (n_{1}+1)(n_{2}+1)-2 $. The proper divisor set of $n$ is

\begin{align*}
D(n)=&\bigg\{p,p^{2},\dots,p^{m_{1}},\dots,p^{n_{1}}, q,q^2,\dots,q^{m_{2}},\dots,q^{n_{2}}, pq, pq^2,\dots,pq^{m_{2}},\dots,pq^{n_{2}}, \dots,\\
&p^{m_{1}}q, p^{m_{1}}q^2,\dots, p^{m_{1}}q^{m_{2}},\dots,p^{m_{1}}q^{n_{2}},\dots,p^{n_{1}}q, p^{n_{1}}q^2,\dots,p^{n_{1}}q^{m_{2}},\dots,p^{n_{1}}q^{n_{2}-1}\bigg\}.
\end{align*}

By the definition of graph $ \mathbb{G}_{n} $, we see that $ p$ is not adjacent to $ p, q,q^{2},\cdots,q^{m_{2}},\cdots,q^{n_{2}} $. So we write adjacency of vertices in terms of iterations and avoid divisors outside the  set $ D(n) $. Thus, we observe that

\begin{align*}
p&\sim p^{i},p^{j}q^{k},~ \text{for}~ i=2,3,\dots,n_{1},~  j=1,2,\dots,n_{1},~ k=1,2,\dots,n_{2},  \\
p^{m_{1}}&\sim p^{i},p^{j}q^{k},~ \text{for}~ i=1,2,\dots,n_{1}, ~i\neq m_{1},~ j=m_{1},\dots,n_{1},~ k=1,2,\dots,n_{2},\\
p^{n_{1}}&\sim p^{i},p^{n_{1}}q^{j},~ \text{for}~ i=1,2,\dots,n_{1}-1,~  j=1,2,\dots,n_{2}-1,\\ % new
q&\sim q^{i},p^{j}q^{k},~ \text{for}~ i=2,3,\dots,n_{2}, ~ j=1,2,\dots,n_{1},~ k=1,2,\dots,n_{2},\\
& \vdots\\
q^{m_{2}}&\sim q^{i},p^{j}q^{k},~ \text{for}~ i=1,2,3,\dots,n_{1}, ~i\neq m_{2}, ~j=1,2,3,\dots,n_{1},~ k=m_{2},\dots,n_{2}\\
q^{n_{2}}&\sim q^{i},p^{j}q^{n_{2}},~ \text{for}~ i=1,2,3,\dots,n_{2}-1,~ j=1,2,3,\dots,n_{1}-1,\\ % % % % % new
pq&\sim p, q, p^{i}q^{j},~ \text{for } ~ i=1,2,3,\dots,n_{1},~ j=1,2,3,\dots,n_{2},\\
& \vdots\\
pq^{m_{2}}&\sim  p, q^{i},pq^{j},p^{k}q^{k} ~ \text{for}~ i=1,2,\dots,m_{2}, ~ j=1,2,3,\dots,n_{2},~ j\neq m_{2},~ k=2,3,\\
pq^{n_{2}}&\sim  p, q^{i}, pq^{j},p^{k}q^{n_{2}}, ~\text{for}~ i=1,2,\dots,n_{2},~ j=1,2,\dots,n_{2}-1,k=2,3,\dots,n_{1}-1,\\ % % % % % new
& \vdots\\
p^{m_{1}}q &\sim  p^{i}, q,p^{m_{1}}q^{j},p^{k}q,p^{l}q^{m} ~\text{for}~ i=1,\dots,m_{1}, ~ j=2,3,\dots,n_{2},~k=1,\dots,m_{1}-1,\\
&l=m_{1}+1,\dots,n_{1},~ m=1,2,\dots,n_{2},\\
& \vdots\\
p^{m_{1}}q^{m_{2}} &\sim  p^{i}, q^{j}, p^{k}q^{l}~ \text{for}~ i=1,2,\dots,m_{1},j=1,2,\dots,m_{2},k=1,2,\dots,n_{1},l=1,\dots,n_{2},\\
p^{m_{1}}q^{n_{2}} &\sim  p^{i}, q^{j},p^{k}q^{n_{2}}, p^{i}q^{j}~ \text{for}~ i=1,2,\dots,m_{1},j=1,2,\dots,n_{2},k=m_{1}+1,\dots,n_{1}-1,\\
p^{n_{1}}q &\sim  p^{i}, q ,p^{j}q, p^{n}q^{k}~ \text{for}~ i=1,2,\dots,n_{1},~j=1,2,\dots,n_{1}-1,~k=2,3\dots,n_{2}-1,\\
p^{n_{1}}q^{m_{2}} &\sim  p^{i}, q^{j} ,p^{n_{1}}q^{k}, p^{i}q^{j}~ \text{for}~ i=1,2,\dots,n_{1},j=1,2,\dots,m_{2},k=m_{2}+1,\dots,n_{2}-1,\\
p^{n_{1}}q^{n_{2}-1} &\sim  p^{i}, q^{j} , p^{i}q^{j}~ \text{for}~ i=1,2,\dots,n_{1},~j=1,2,\dots,n_{2}-1.\\
\end{align*}

Therefore, by Theorem <ref>, we have
\begin{align*}
\mathcal{P}(\mathbb{Z}_{n})= &K_{\phi(n)+1}\triangledown \mathbb{G}_{n}[K_{\phi(p)},\dots,K_{\phi(p^{m_{1}})},\dots,K_{\phi(p^{n_{1}})},K_{\phi(q)},\dots,K_{\phi(q^{m_{2}})},\dots,K_{\phi(q^{n_{2}})},\\
\end{align*}

Now, by using Theorem <ref>, we calculate the values of $ \alpha_{i} $'s and $ r_{i}+\alpha_{i}=r_{i}^{'} $'s. We recall that [102] $\phi(xy)=\phi(x)\phi(y),$ provided that $(x,y)=1$, $\sum\limits_{i=1}^{k}\phi(p^i)=p^k-1$ and $ \sum\limits_{d|s}\phi(d)=s $. Using this information and definition of $ \alpha_{i} $'s, we have
\[
\alpha_{1}=\sum\limits_{1,n\neq d|n}\phi(d)=n-1-\phi(n). \]

\begin{align*}
\alpha_{2}=& \phi(p^{2})+\dots+\phi(p^{m_{1}})+\dots+\phi(p^{n_{1}})+\phi(pq)+\dots+\phi(pq^{m_{2}})+\dots\phi(pq^{n_{2}})\\
=&\sum\limits_{1,p,n\neq d|n}\phi(d)-[\phi(q)+\dots+\phi(q^{m_{1}})+\dots+\phi(q^{n_{1}})]+\phi(n)+1\\
\end{align*}
Proceeding in the same way as above, other $ \alpha_{i} $'s are

\begin{align*}
\alpha_{3}=&q^{n_{2}}(p^{n_{1}}-p)+p-\phi(p^{2}),\\
& \vdots\\
\alpha_{m_{1}+1}=& p^{m_{1}-1}+q^{n_{2}}(p^{n_{1}}-p^{m_{1}-1}) -\phi(p^{m_{1}}),\\
\vdots\\
\alpha_{n_{1}+1}=&p^{n_{1}-1}+\phi(p^{n_{1}})(q^{n_{2}-1}-1),~\alpha_{n_{1}+2} =n-\phi(q)-p^{n_{1}}+1,\\
\alpha_{n_{1}+m_{1}+1}=&q^{m_{2}-1}+p^{n_{1}}(q^{n_{2}}-q^{m_{2}-1})-\phi(q^{m_{2}}),\\
\vdots\\
\alpha_{n_{1}+n_{2}+1} =&q^{n_{2}-1}+\phi(q^{n_{2}})(p^{n_{1}}-1),\\
\alpha_{n_{1}+n_{2}+2}=&\phi(p)+\phi(q)+1-\phi(pq)+(q^{n_{2}}-1)(p^{n_{1}}-1),\\
\alpha_{n_{1}+n_{2}+m_{1}+1}=&q^{n_{2}}(p^{n_{1}}-1)+q^{m_{2}}-q^{m_{2}-1}(p^{n_{1}}-p)-\phi(pq^{m_{2}}),\\
\alpha_{2n_{1}+n_{2}+1}=&pq^{n_{2}}-\phi(pq^{n_{2}})+\phi(p^{n_{1}})(q^{n_{2}}-q),\\
\alpha_{m_{1}n_{2}+n_{1}+2}=&p^{m_{1}}-\phi(p^{m_{1}}q)+p^{n_{1}}(q^{n_{2}}-1) -p^{m_{1}-1} (q^{v_{1}}-q),\\
\alpha_{m_{1}n_{2}+n_{1}+m_{1}+1}=&n+p^{m_{1}-1}q^{m_{2}-1}+p^{m_{1}}q^{m_{2}}-2\phi(p^{m_{1}}q^{m_{2}})-q^{n_{2}}p^{m_{1}-1}-p^{n_{2}}q^{m_{2}-1},\\
\alpha_{(m_{1}+1)n_{2}+n_{1}+1}=&p^{m_{1}}q^{n_{2}}-\phi(p^{m_{1}}q^{n_{2}})+\phi(q^{n_{2}})(p^{n_{1}}-p^{m_{1}}),\\
\alpha_{n_{1}n_{2}+n_{1}+2}=&p^{n_{1}}q+\phi(p^{n_{1}})(q^{n_{2}}-q)-\phi(p^{n_{1}}q),\\
\alpha_{n_{1}n_{2}+n_{1}+m_{1}+1}=&p^{n_{1}}q^{m_{2}}+\phi(p^{n_{1}})(q^{n_{2}}-q^{m_{2}})-\phi(p^{n_{1}}q^{m_{2}}),\\
\alpha_{(n_{1}+1)n_{2}+n_{1}}=&p^{n_{1}}q^{n_{2}-1}+\phi(n)-\phi(p^{n_{1}}q^{n_{2}-1}).
\end{align*}

Also, value of $r_{i}+\alpha_{i}= r_{i}^{'} $'s are given by

\begin{align*}
r_{1}^{'}=&n-1,~ r_{2}^{'}=n-q^{n_{2}},\\
r_{n_{1}+1}^{'}=&p^{n_{1}-1}+\phi(p^{n_{1}})q^{n_{2}}-1, ~r_{n_{1}+2}^{'}=n-p^{n_{1}},\\
r_{m_{1}n_{2}+n_{1}+2}^{'}=&p^{m_{1}}+p^{n_{1}}(q^{n_{2}}-1) -p^{m_{1}-1} (q^{n_{1}}-q)-1,\\
r_{m_{1}N_{2}+N_{1}+m_{1}+1}^{'} =&n+p^{m_{1}-1}q^{m_{2}-1}+p^{m_{1}}q^{m_{2}}-\phi(p^{m_{1}}q^{m_{2}})-q^{n_{2}}p^{m_{1}-1}-p^{n_{2}}q^{m_{2}-1}-1,\\
r_{n_{1}n_{2}+n_{1}+m_{1}+1}^{'} =&p^{n_{1}}q^{n_{2}-1}+\phi(n)-1.
\end{align*}

By  Theorem <ref>, $ \dfrac{n}{n-1} $ is the normalized Laplacian eigenvalue of $ \mathcal{P}(\mathbb{Z}_{n}) $ with multiplicity $ \phi(n). $  We note that each of $ G_{i}=K_{i} $ and by Theorem <ref>, we see that

\[ \dfrac{\phi(d_{1})+\alpha_{2}}{\phi(d_{1})+\alpha_{2}-1}=\dfrac{\phi(p)+n-\phi(p)-q^{n_{2}}+1}{\phi(p)+n-\phi(p)-q^{n_{2}}+1-1}=\dfrac{n-q^{n_{2}}+1}{n-q^{n_{2}}}, \]
is the normalized Laplacian eigenvalue of $ \mathcal{P}(\mathbb{Z}_{n}) $ with multiplicity $ \phi(p)-1 $. Similarly, other normalized Laplacian eigenvalues of $ \mathcal{P}(\mathbb{Z}_{n}) $ can be found. Also, substituting the values of $ \alpha_{i} $'s, $ r_{i}^{'} $'s and using the adjacency relations, the remaining normalized Laplacian eigenvalues of $ \mathcal{P}(\mathbb{Z}_{n}) $ are the eigenvalues of matrix (<ref>).

CHAPTER: DISTANCE SIGNLESS LAPLACIAN EIGENVALUES  AND GENERALIZED DISTANCE ENERGY OF GRAPHS

In this chapter, we obtain the distance signless Laplacian eigenvalues of the joined union of regular graphs. We also find the distance signless Laplacian eigenvalues of the zero-divisor graphs associated to ring $ \mathbb{Z}_{n} $. Also, we define the generalized distance energy and obtain the bounds for it. We characterize the graphs attaining these bounds. We show that the complete bipartite graph and the star graph have the minimum generalized distance energy among the class of bipartite graphs and trees respectively.

§  INTRODUCTION

In a graph $G$, the distance between any two vertices $u,v\in V(G),$ denoted by $d(u,v)$, is defined as the length of a shortest path between $u$ and $v$. The distance matrix of $G$, denoted by $\mathcal{D}(G)$, is defined as $\mathcal{D}(G)=(d_{uv})$, where $u,v\in V(G)$. More about distance matrix can be found in the survey paper [6]. The transmission degree $Tr_{G}(v)$ of a vertex $v$ is defined to be the sum of the distances from $v$ to all other vertices in $G$, that is, $Tr_{G}(v)=\sum\limits_{u\in V(G)}d(u,v)$. A graph $ G $ is said to be $k$-transmission regular if $ Tr_{G}(v)=k,$ for each $ v\in V(G). $  The transmission number or Wiener index of a graph $ G, $ denoted by $W(G), $ is the sum of distances between all unordered pairs of vertices in $G$.
It is clear that $W(G)=\frac{1}{2}\displaystyle\sum_{v\in V(G)}Tr_{G}(v)$. If $Tr_G(v_i)$ (or simply $Tr_{i}$) is the transmission degree of the vertex $v_i\in V(G)$, the sequence $\{Tr_{1},Tr_{2},\ldots,Tr_{n}\}$ is called the transmission degree sequence of  $G$.

Let $Tr(G)=diag (Tr_1,Tr_2,\ldots,Tr_n) $ be the diagonal matrix of vertex transmissions of $G$. Aouchiche and Hansen [5] introduced the Laplacian and the signless Laplacian for the distance matrix of a connected graph. The matrix $ \mathcal{D}^L(G)=Tr(G)-\mathcal{D}(G) $ is called the distance Laplacian matrix of $G$, while the matrix  $\mathcal{D}^{Q}(G)=Tr(G)+\mathcal{D}(G)$ is called the distance signless Laplacian matrix of $G$. The matrices $ \mathcal{D}(G) $ and $\mathcal{D}^{Q}(G)$ are real symmetric and positive definite for $ n\geq 3 $, so their eigenvalues can be arranged as $ \rho^{\mathcal{D}}_n(G)\leq \ldots\leq\rho^{\mathcal{D}}_2(G)\leq \rho^{\mathcal{D}}_1(G) $ and  $\rho^{Q}_n(G)\leq \ldots\leq\rho^{Q}_2(G)\leq \rho^{Q}_1(G)$, respectively.  The eigenvalues $ \rho^{\mathcal{D}}_1(G) $ and $ \rho^{Q}_1(G) $ are called the distance  and distance signless Laplacian spectral radii of $ G. $ Since the matrix $\mathcal{D}^{Q}(G)$ is non-negative and irreducible, so by Perron-Frobenius Theorem, $ \rho^{Q}_1(G) $ is positive and simple zero of the characteristic polynomial of $\mathcal{D}^{Q}(G)$. There corresponds a unique eigenvector with positive entries having unit length, which is called the distance signless Laplacian Perron vector of graph $ G. $ More work on distance Laplacian and distance signless Laplacian matrix can be found in [7, 8, 38, 56, 61, 106, 107, 129, 130] and references therein.

§ DISTANCE SIGNLESS LAPLACIAN EIGENVALUES OF THE JOINED UNION OF REGULAR GRAPHS

We compute the distance signless Laplacian spectrum of the joined union of graphs $G_1,G_2,\dots,G_n$ in terms of the adjacency spectrum of the graphs $G_1,G_2,\dots,G_n$ and the eigenvalues of quotient matrix.

Let $G$ be a graph of order $n$ having vertex set $V(G)=\{ v_{1}, \ldots, v_{n}\}.$ Let $ G_{i}$ be $r_{i}$-regular graphs of order $ n_{i} $ having adjacency eigenvalues $\lambda_{i1}=r_{i}\geq \lambda_{i2}\geq\ldots\geq \lambda_{in_{i}}, $ where $ i=1,2, \ldots, n$.  The distance signless Laplacian spectrum of the joined union graph $ G[G_{1},\ldots, G_{n}] $ of order $ \sum\limits_{i=1}^{n}n_{i} $ consists of the eigenvalues $ 2n_i+n_{i}^{\prime}-r_{i}-\lambda_{ik}-4$ for $ i=1,\ldots,n $ and $ k=2,3,\ldots, n_{i} $, where $n_{i}^{\prime}=\sum\limits_{k=1,k\ne i}^{n}n_kd_{G}(v_i,v_k)$. The remaining $n$ eigenvalues are given by the equitable quotient matrix

\begin{equation*}
\mathcal{Q}=\begin{pmatrix}
4n_{1}+n_{1}^{'}-2r_{1}-4& n_{2}d_{G}(v_{1}, v_{2})&\ldots & n_{n}d_{G}(v_{1},v_{n})\\
n_{1}d_{G}(v_{2}, v_{1})&4n_{2}+n_{2}^{'}-2r_{2}-4& \ldots & n_{n}d_{G}(v_{2}, v_{n})\\
\vdots &\vdots  &\ddots &\vdots\\
n_{1}d_{G}(v_{n}, v_{1})& n_{2}d_{G}(v_{n}, v_{2})&\ldots &4n_{n}+n_{n}^{'}-2r_{n}-4
\end{pmatrix}.
\end{equation*}

Let $G$ be a connected  graph of order $n$ having vertex set $V(G)=\{ v_{1}, \ldots, v_{n}\}$ and let $V(G_i)=\{ v_{i1}, \ldots, v_{in_i}\}$ be the vertex set of the graph $G_i$, for $i=1,2,\dots,n$. Let  $H=G[G_{1}, \ldots, G_{n}]$ be the joined union  of the graphs $G_1,G_2,\dots, G_n$. The transmission degree of each vertex $v_{ij}\in V(H)$, for $1\le i\leq n$ and $1\leq j\leq n_i$, is given by

\begin{equation}\label{transmission}
\end{equation}

where $n_{i}^{'}=\sum\limits_{k=1}^{n}n_kd_{G}(v_i,v_k)$ is same for every vertex belonging to $ V(G_i)$.
We note that
$ Tr(v_{i,j})$ is equal to the distance of $j$th vertex in $G_{i} $ to all other vertices of $ G$, which is further equal to the distance within vertices of $ G_{i} $ and the distance of vertices outside of  $ G_{i} $. So

\begin{align*}
Tr(v_{i,j})&= d(v_{i,j})+2(n_{i}-1-d(v_{i,j})) + n_{1}d(v_{i},v_{1})+n_{2}d(v_{i},v_{2})+\dots+n_{n}d(v_{i},v_{n})\\&=2n_{i}-2-d(v_{i,j})+n_{i}^{'},
\end{align*}

and thus Equation (<ref>) is true for any diameter.
Under the suitably labelling of the vertices of $H$, the distance signless Laplacian matrix $D^{Q}(H)$ can be written as

\begin{equation*}
\mathcal{D}^{Q}(H)=\begin{pmatrix}
h_1&d_{G}(v_{1}, v_{2})J_{n_1\times n_2}& \ldots & d_{G}(v_{1}, v_{n})J_{n_1\times n_n} \\
d_{G}(v_{2}, v_{1})J_{n_2\times n_1}&h_2& \ldots & d_{G}(v_{2}, v_{n})J_{n_2\times n_n}\\
\vdots &\vdots &\ddots &\vdots\\
d_{G}(v_{n}, v_{1})J_{n_n\times n_1}& d_{G}(v_{n}, v_{2})J_{n_n\times n_{n-1}}&\ldots &h_n
\end{pmatrix},
\end{equation*}

where,  for $i=1,2,\ldots,n,$

\begin{equation*}
\end{equation*}

in which, for $ i=1,2,\ldots,n,$
$A(G_i)$ is the adjacency matrix of graph $ G_{i}$, and $J_{n_i}$ is the matrix having all entries $ 1 $ and $ I_{n_i}$ is the identity matrix of order $n_i$.

As $ G_{i} $ is an $ r_{i} $ regular graph, so $e_{n_i}=(1,1,\dots,1)^T$ with $ n_{i} $ entries is the eigenvector of the adjacency matrix $ A(G_{i}) $ corresponding to the eigenvalue $r_{i} $ and all other eigenvectors are orthogonal to $e_{n_i}.$ Let $ \lambda_{ik}$, $2\leq k\leq n_i$, be any eigenvalue of $ A(G_{i})$ with the corresponding eigenvector $X=(x_{i1},x_{i2},\dots,x_{in_i})^T$ satisfying $e_{n_i}^TX=0.$ It is a well known fact that the column vector $X$ can be regarded as a function defined on $ V(G_i) $ relating vertex $ v_{ij} $ to $ x_{ij} $, that is, $ X(v_{ij})=x_{ij} $ for $ i=1,2,\ldots,n $ and $j=1,2,\dots,n_i$. Now, consider the vector $Y=(y_{1},y_{2},\dots,y_{n})^T$, where

\begin{equation*}
\begin{array}{rl}
x_{ij}&~\text{if}~ v_{ij}\in V(G_i)\\
\end{array}\right.
\end{equation*}

Noting that $ e_{n_{i}}^{T}X=0 $ and coordinates of the  vector $Y$ corresponding to vertices in $\cup_{j\ne i}V_j$ of $H$ are zeros, therefore, we have

\begin{align*}
\vdots\\
(2n_i+n_{i}^{'}-d(v_{ij})-4)X-\lambda_{ik} X\\
\vdots\\
\end{pmatrix}=(2n_i+n_{i}^{'}-d(v_{ij})-\lambda_{ik}-4)Y.
\end{align*}

This shows that the vector $Y$ is an eigenvector of $\mathcal{D}^{Q}(H) $ corresponding to the eigenvalue $2n_i+n_{i}^{\prime}-d(v_{ij})-\lambda_{ik}-4$, for every eigenvalue $\lambda_{ik}$, $2\leq k\leq n_i$, of $A(G_i)$. It follows that $2n_i+n_{i}^{\prime}-r_{i}-\lambda_{ik}-4$, for $1\leq i\leq n$ and $2\leq k\leq n_i$, is an eigenvalue of $ \mathcal{D}^{Q}(H) $ and in this way we have obtained $ \sum\limits_{i=1}^{n}n_{i}-n$  eigenvalues. The remaining $ n $ eigenvalues are the zeros of the characteristic polynomial of the following equitable quotient matrix.

\begin{align*}
\begin{pmatrix}
4n_{1}+n_{1}^{'}-2r_{1}-4& n_{2}d_{G}(v_{1}, v_{2})&\ldots & n_{n}d_{G}(v_{1},v_{n})\\
n_{1}d_{G}(v_{2}, v_{1})&4n_{2}+n_{2}^{'}-2r_{2}-4& \ldots & n_{n}d_{G}(v_{2}, v_{n})\\
\vdots &\vdots  &\ddots &\vdots\\
n_{1}d_{G}(v_{n}, v_{1})& n_{2}d_{G}(v_{n}, v_{2})&\ldots &4n_{n}+n_{n}^{'}-2r_{n}-4
\end{pmatrix}.
\end{align*}
This completes the proof.

The following consequence of Theorem <ref> gives the distance signless Laplacian spectrum of the join of two regular graphs, in which one of the graphs is the union of two regular graphs having distinct vertex sets.

For $ i=1,2,3$, let $ G_{i} $ be $ r_{i} $ regular graphs of orders $ n_{i} $ having adjacency eigenvalues $ \lambda_{i,1}=r_{i}\geq \lambda_{i,2}\geq \dots\geq\lambda_{i,n_{i}} $.  Then the distance signless Laplacian spectrum of $ G_{1}\triangledown (G_{2}\cup G_{3}) $ is

\begin{align*}
\Big\{ (n+n_{1}-r_{1}-\lambda_{1,k}-4)^{[n_{1}-1]}, (2n-n_{1}-&r_{2}-\lambda_{2,k}-4)^{[n_{2}-1]},\\ &(2n-n_{1}-r_{3}-\lambda_{3,k}-4)^{[n_{3}-1]} \Big\},
\end{align*}

where $ k=2,3,\dots,n_{i} $, for $ i=1,2,3 $ and $ n=n_{1}+n_{2}+n_{3} .$ The remaining three eigenvalues are given by the equitable quotient matrix

\begin{equation*}
\begin{pmatrix}
n+3n_{1}-2r_{1}-4 &n_{2} &n_{3}\\
n_{1} & 2n+2n_{2}-n_{1}-2r_{2}-4 &2n_{3}\\
n_{1} &2n_{2} & 2n+2n_{3}-n_{1}-2r_{3}-4
\end{pmatrix}.
\end{equation*}

The next consequence gives the distance signless Laplacian spectrum of the join of two regular graphs.

Let $ G_{1} $ and $ G_{2} $ be $ r_{1} $ and $ r_{2} $ regular graphs of order $ n_{1} $ and $ n_{2} $, respectively. Let $ \lambda_{1}=r_{1},\lambda_{2}, \dots, \lambda_{n_{1}} $ and $ \lambda^{'}_{1}=r_{2},\lambda^{'}_{2},\dots,\lambda^{'}_{n_2} $ be the adjacency eigenvalues of $ G_{1} $ and $ G_{2} $, respectively. Then the distance signless Laplacian spectrum of $ G_{1}\triangledown G_{2} $ consists of eigenvalues $ 2n-n_{2}-r_{1}-4-\lambda_i $ and $ 2n-n_{1}-r_{2}-4-\lambda^{'}_j $ with multiplicities $ n_{1}-1 $ and $ n_{2}-1 $ respectively, where $n=n_1+n_2,~2\leq i\leq n_1$ and $2\leq j\leq n_2$. The remaining two eigenvalues are given by the eigenvalues of the equitable quotient matrix

\begin{equation} \label{eq21}
\begin{pmatrix}
n+3n_{1}-2r_{1}-4& n_{2}\\
n_{1}& n+3n_{2}-2r_{2}-4

\end{pmatrix}.
\end{equation}

The following are immediate consequences of Corollary <ref> and give the distance signless Laplacian eigenvalues of a complete bipartite and a complete split graph.

The distance signless Laplacian spectrum of $ K_{a,b} $ consists of the eigenvalue $ 2a+b-4 $ with multiplicity $ a-1$, the eigenvalue $2b+a-4$ with multiplicity $ b-1 $ and the eigenvalues

\begin{equation*}
\dfrac{1}{2}\left ( 5(a+b)-8\pm \sqrt{9(a-b)^2+4ab} \right ).
\end{equation*}

The distance signless Laplacian eigenvalues of $CS_{\omega,n-\omega}$ are given by
\[ \left\lbrace  (n-2)^{[\omega-1]}, (2n-\omega-4)^{[n-\omega-1]}, \frac{1}{2}\left(5n-2\omega-6\pm \sqrt{D}\right)\right\rbrace,  \]

where $ (3(2\omega-n)-2(\omega-1))^{2}+4\omega(n-\omega). $

§ DISTANCE SIGNLESS LAPLACIAN EIGENVALUES OF ZERO-DIVISOR GRAPHS ASSOCIATE TO THE RING $ \MATHBB{Z}_{N} $

For a commutative ring $R$ with identity denoted by $1$, let the set $Z(R)$ denote the set of zero-divisors and let $Z^{*}(R)=Z(R)\setminus \{0\}$ be the set of non-zero zero-divisors of $R$.  The zero-divisor graph of $R$, denoted by $\Gamma(R)$, is a simple graph whose vertex set is $Z^{*}(R)$ and two vertices $u, v \in Z^*(R)$ are adjacent if and only if $uv=vu=0$.
The zero-divisor graphs of commutative rings were first introduced by Beck [14]. In the definition he included the additive identity and was interested mainly in coloring of commutative rings. Later Anderson and Livingston [4] modified the definition of zero-divisor graphs and excluded the additive identity of the ring in the zero-divisor set. We denote the ring of integers modulo $n$ by $\mathbb{Z}_n$. The order of the zero-divisor graph $\Gamma(\mathbb{Z}_n)$ is $ n-\phi(n)-1 $, where $ \phi $ is Euler's totient function. The zero-divisor graphs have been extensively studied in [3, 4, 14, 124]. The spectral properties of adjacency and Laplacian  matrices for zero-divisor graphs were discussed in [26, 109, 145].

For integers $d$ and $n$ with $ 1<d<n$, if $ d $ is a proper divisor of $ n $, we write $ d|n $. Let $ d_{1}, d_{2},\dots,d_{t} $ be the distinct proper divisors of $ n$. Let $ \Upsilon_{n} $ be the simple graph with vertex set $ \{d_{1}, d_{2},\dots,d_{t}\} $, in which two distinct vertices are adjacent if and only if $ n|d_{i}d_{j}$. Let the prime factorization of $n$ be $ n=p_{1}^{n_{1}}p_{2}^{n_{2}}\dots p_{r}^{n_{r}} $, where $ r,n_{1},n_{2},\dots,n_{r} $ are positive integers and $ p_{1},p_{2},\dots,p_{r} $ are distinct prime numbers. The order of $ \Upsilon_{n} $ is given by

\begin{equation*}
|V( \Upsilon_{n} )|=\prod\limits_{i=1}^{r}(n_{i}+1)-2.
\end{equation*}

This $  \Upsilon_{n}  $ is connected (see [26]) and plays a fundamental role in the present section.
For $ 1\leq i \leq t $, we consider the sets $ A_{d_{i}}= \{ x\in \mathbb{Z}_{n} : (x,n)=d_{i} \}$, where $(x,n)$ denotes the greatest common divisor of $x$ and $n$. We see that $ A_{d_{i}} \cap A_{d_{j}}=\phi$, when $ i\neq j $, implying that the sets $ A_{d_{1}}, A_{d_{2}}, \dots, A_{d_{t}} $ are pairwise disjoint and partitions the vertex set of $ \Gamma(\mathbb{Z}_{n}) $ as $V(\Gamma(\mathbb{Z}_{n}))=A_{d_{1}}\cup A_{d_{2}}\cup \dots \cup A_{d_{t}}.$ From the definition of $ A_{d_{i}} $, a vertex of $ A_{d_{i}} $ is adjacent to the vertex of $ A_{d_{j}} $ in $ \Gamma(\mathbb{Z}_{n}) $ if and only if $ n $ divides $ d_{i}d_{j} $ , for $ i,j\in \{1,2,\dots,t\} $ (see [26]).

We have the following observation.

If $ d_{i} $ is the proper divisor of $ n $, then
$|A_{d_{i}}|=\phi\left( \frac{n}{d_{i}}\right)$, for $ 1\leq i \leq t .$

The induced subgraphs $ \Gamma(A_{d_i}) $ of $ \Gamma(\mathbb{Z}_{n}) $ are either cliques or their complements as can be seen in the following lemma.

Let $ n $ be a positive integer and $ d_{i} $ be its proper divisor. Then following hold.

(i) For $i\in\{1,2,\dots,t\} , $ the induced subgraph $ \Gamma(A_{d_i}) $ of $ \Gamma(\mathbb{Z}_{n}) $ on the vertex set $ A_{d_{i}} $ is either the complete graph $ K_{\phi\left (\frac{n}{d_{i}}\right )} $ or its complement $\overline{ K}_{\phi\left (\frac{n}{d_{i}}\right )} .$ Indeed, $ \Gamma(A_{d_i}) $ is $ K_{\phi\left (\frac{n}{d_{i}}\right )} $ if and only $ n $ divides $ d_{i}^{2} .$

(ii) For $i,j\in\{1,2,\dots,t\} $ with $ i\neq j ,$ a vertex of $ A_{d_{i}} $ is adjacent to either all or none of the vertices in $ A_{d_{j}} $ of $ \Gamma(\mathbb{Z}_{n}) $.

The following lemma says that $ \Gamma(\mathbb{Z}_{n}) $ is the joined union of certain complete graphs and null graphs.
[26] Let $ \Gamma(A_{d_i}) $ be the induced subgraph of $ \Gamma(\mathbb{Z}_{n}) $ on the vertex set $ A_{d_i} $ for $ 1\leq i\leq t .$ Then $ \Gamma(\mathbb{Z}_{n}) = \Upsilon_{n}[\Gamma(A_{d_1}),\Gamma(A_{d_2}),\dots,\Gamma(A_{d_t})]. $

In the following example, we illustrate the structure of $ \Gamma(\mathbb{Z}_{30}) $ and find its distance signless Laplacian eigenvalues.

[fill=black] (0,0) circle (2pt);[fill=black] (2,0) circle (2pt);[fill=black] (1,1.3) circle (2pt);
[fill=black] (1,2.3) circle (2pt);[fill=black] (-1,0) circle (2pt);[fill=black] (3,0) circle (2pt);

[thin] (0,0)–(2,0); [thin] (0,0)–(1,1.3);[thin] (2,0)–(1,1.3);
[thin] (0,0)–(-1,0); [thin] (2,0)–(3,0);[thin] (1,1.3)–(1,2.3);

at (-1,-.3) $ 3 $;at (0,-.3) $ 10 $;
at (1.35,1.3) $ 15 $;at (1.25,2.3) $ 2 $;
at (3,-.3) $ 5 $;at (2,-.3) $ 6 $;

[scale=.9, auto=left, every node/.style=circle, fill=blue!10, minimum width=0.1cm]
(n1) at (0,0) $ \overline{K}_{2} $;(1) at (-2,0) $ \overline{K}_{4} $;
(n2) at (2,0) $ \overline{K}_{4} $;(4) at (4,0) $ \overline{K}_{2} $;
(n3) at (1,2) $ \overline{K}_{1} $;(3) at (1,4) $ \overline{K}_{8} $;

/→in n1/n2,n1/n3,n2/n3, n2/4, n1/1, n3/3

Proper divisor graph $ \Upsilon_{30} $ and zero divisor graph $  \Gamma(\mathbb{Z}_{30})  $.

Example Let $ n=30 $. Then $ 2,3,5,6,10 $ and $ 15 $ are the proper divisors of $ n $ and $ \Upsilon_{n} $ is the graph $ G_{6} : 3\sim 10\sim 6 \sim 5, 10\sim 15\sim 2 $ and $ 6\sim 15 $, that is $ \Upsilon_{n} $ is the triangle graph having a pendent edge at each vertex of the triangle. Applying Lemma <ref>, we have

\begin{equation*}
\Gamma(\mathbb{Z}_{30})=\Upsilon_{30}[\overline{K}_{4},\overline{K}_{2},\overline{K}_{4},\overline{K}_{2},K_{1},\overline{K}_{8}].
\end{equation*}

By Theorem <ref>, the distance signless Laplacian spectrum of $ \Gamma(\mathbb{Z}_{30}) $ consists of the eigenvalues $ \{46^{[4]}, 43^{[7]}, 33^{[3]}, 29\} $, and the remaining eigenvalues are given by

\begin{equation*}
\begin{pmatrix}
54 & 2 & 8 & 6 & 2 & 24\\
4 & 33 & 4 & 4 & 1 & 16\\
8 & 2 & 41 & 2 & 1 & 16\\
8 & 4 & 4 & 50 & 2 & 24\\
8 & 2 & 4 & 4& 26 & 8\\
12 & 4 & 8 & 6 & 1 & 59

\end{pmatrix}.
\end{equation*}

The characteristic polynomial of the above matrix is $ x^6-263x^{5}+27575x^{4}-1489941x^{3}+44016528x^{2}-676666908x+4239569664 $
and it's approximated zeros are
\begin{equation*} \{87.1555, 44.5461, 40.5727, 35.0098, 30.5597, 25.1562\}. \end{equation*}

Next we will discuss the distance signless laplacian spectrum of $ \Gamma(\mathbb{Z}_n) $ for $ n\in\{pq,p^{2}q,p^3,p^4\} $ with the help of Theorem <ref>. Let $ n=pq $, where $ p $  and $ q $, $ p<q $ are distinct primes. Then  by Lemma <ref> and Lemma <ref>, we have

\begin{equation}\label{eq pq}
\Gamma(\mathbb{Z}_{pq})= \Upsilon_{pq}[\Gamma(A_{p}),\Gamma(A_{q})] =K_{2}[\overline{K}_{\phi(p)},\overline{K}_{\phi(q)}]=\overline{K}_{\phi(p)}\triangledown \overline{K}_{\phi(q)}=K_{\phi(p),\phi(q)}.\end{equation}

The following result gives the distance signless Laplacian eigenvalues of $ \Gamma(\mathbb{Z}_{n}) $, when $ n $ is product of two primes.

The distance signless Laplacian eigenvalues of $ \Gamma(\mathbb{Z}_{n}) $ are $ 2n-q-3 $ and $ 2n-p-3 $ with multiplicities $ p-2 $ and $ q-2 $ respectively, and

\begin{equation*}
\frac{1}{2}\left( 5n-8\pm\sqrt{9(p-q)^{2}+4(p-1)(q-1)}\right). \end{equation*}

Proof. Let $ n=pq $, where $ p<q $ are distinct primes. Then the proper divisors of $ n $ are $ p $ and $ q $, so $ \Upsilon_{pq} $ is $ K_{2} $. By Equation (<ref>) and Theorem <ref>, the  result follows.

Next, result gives the distance signless Laplacian eigenvalues of $\Gamma(\mathbb{Z}_{n}) $ for $ n=p^{2}q $.

The distance signless Laplacian spectrum of $ \Gamma(\mathbb{Z}_{p^{2}q}) $ is
\begin{align*}
\{(2pq+3p^2-4p-5)^{[pq-p-q]},&(pq+2p^2-2p-3)^{[p-2]},(2pq+p^2-2p-4)^{[q-2]},\\& (3pq+2p^2-3p-2q-4)^{p^2-p-1},x_{1}\geq x_{2} \geq x_{3} \geq x_{4}\} \end{align*}
where $ x_{1}, x_{2}, x_{3}, x_{4} $ are the zeros of the characteristic polynomial of the matrix.

Proof. Let $ n=p^{2}q, $ where $ p $ and $ q $ are distinct primes. Since proper divisors of $ n $ are $ p, q, pq, p^2 $, so $ \Upsilon_{p^{2}q} $ is the path $ P_{4} : q\sim p^2 \sim pq\sim p$. By Lemma <ref>, we have
\begin{equation*}
\Gamma(\mathbb{Z}_{p^2q})= \Upsilon_{p^2q}[\Gamma(A_{q}),\Gamma(A_{P^2}),\Gamma(A_{pq}),\Gamma(A_{p})] =P_{4}[\overline{K}_{\phi(p^2)},\overline{K}_{\phi(q)},K_{\phi(p)},\overline{K}_{\phi(pq)}] .\end{equation*}

Now, by Theorem <ref>, clearly $ 2pq+3p^2-4p-5,pq+2p^2-2p-3,2pq+p^2-2p-4$ and $3pq+2p^2-3p-2q-4$ are the distance signless Laplacian eigenvalues with multiplicities $pq-p-q, p-2, q-2$ and $ p^2-p-1$ respectively. The remaining four distance signless Laplacian eigenvalues are given by the equitable quotient matrix

\begin{equation*}
\begin{pmatrix}
p(4p+3q-5)-2q-4& p-1&2(q-1)& 3p(p-1)\\
pq-p-q+1 & p(p+2q-2)+2q-7 & q-1 & 2p(p-1)\\
2(pq-p-q+1) & p-1& p(2p+q-1)-4 & p(p-1)\\
3(pq-p-q+1)& 2(p-1)& q-1&d_{4}

\end{pmatrix},
\end{equation*}

where $ d_{4}=p(3p+4q-6)-2q-3 $

A graph $G$ is called distance signless Laplacian integral graph if all its distance signless Laplacian eigenvalues are integers. In order to see when the joined graph or a zero divisor graph $ \Gamma(\mathbb{Z}_n) $ is distance signless Laplacian integral, we have the following proposition.

The zero divisor graph $ \Gamma(\mathbb{Z}_n) $ is distance signless Laplacian integral if and only if matrix $ Q $ of Theorem <ref> is integral.

The following result say that $ \Gamma(\mathbb{Z}_{n}) $ is distance singnless Laplacian integral only for prime power.

Let $ n=p^2 $, where $ p $ is any prime. Then  $ \Gamma(\mathbb{Z}_{n}) $  is distance signless Laplacian integral and  spectrum is given by $ \{2p-4, (p-3)^{[p-2]}\} .$

Proof. Since $ \Gamma(\mathbb{Z}_{p^2}) =\Gamma(A_{p})$ is the complete graph $ K_{p-1} ,$ the result follows according to $ p=2 $ or $ p>2 .$

In the next result, we find distance signless Laplacian eigenvalues of the complete split graph  of $ \Gamma(\mathbb{Z}_{n}). $

Let $n=p^3  $. Then the distance signless Laplacian eigenvalues of $ \Gamma(\mathbb{Z}_{n}) $ are $ 2n-p-3 $ and $ 2n-p^2-1 $ with multiplicities $ p^2-p-1 $ and $ p-2 $ respectively, and

\begin{equation*}
\left\lbrace \frac{1}{2}\left(5n-2(p-1)-6\pm \sqrt{(3(2p-2-n)-2(p-2))^{2}+4(p-1)(n-p+1)}\right)\right\rbrace. \end{equation*}

Proof. Since proper divisors of $ n $ are $ p $ and $ p^2 $, so $ \Upsilon_{n} $ is $ K_{2}: $ $ p\sim p^2 $. By Lemma <ref>, we have

\begin{equation*}
\Gamma(\mathbb{Z}_{p^3}) =\Upsilon_{p^3}[\Gamma(A_{p}),\Gamma(A_{p^2})] =K_{2}[\overline{K}_{\phi(p^2)},\overline{K}_{\phi(p)}]=\overline{K}_{p(p-1)}\triangledown K_{p-1} .\end{equation*}

This implies that $ \Gamma(\mathbb{Z}_{p^3}) $ is a complete split graph of order $ p^2-1 $, having independent set of cardinality $ p(p-1) $ and clique of order $ p-1 $. By Theorem <ref> and Corollary <ref>, the result follows.

Next, proposition gives the distance signless Laplacian eigenvalues of $ \Gamma(\mathbb{Z}_{p^{4}}) $.

The distance signless Laplacian eigenvalues of $ \Gamma(\mathbb{Z}_{n}) $, where $ n=p^4 $, are $ n-2 $, $ 2n-p-3$ and $ 2n-p^2-1 $ with multiplicities $ p-2 $,  $ p^2(p-1)-1 $ and $ p(p-1)-1 $ respectively and three more distance signless Laplacian eigenvalues given by

\begin{equation*}
\begin{pmatrix}
n+p-3 & p^2(p-1) & p(p-1)\\
p-1 & 2n+p(2p^2-2p-1)-3 & 2p(p-1)\\
p-1 & 2p^2(p-1) & 2n-p-1
\end{pmatrix}. \end{equation*}

As proper divisors of $ n $ are $ p, p^2 $ and $ p^3 $, so $ \Upsilon_{n} $ is $ P_{3}: $ $ p\sim p^3\sim p^2 $. By Lemmas <ref>, <ref> and <ref>, we have
$ \Gamma(A_{p})=\overline{K}_{\phi(p^3)}=\overline{K}_{p^2(p-1)}, \Gamma(A_{p^2})=K_{\phi(p^2)}=K_{p(p-1)}, $ and $ \Gamma(A_{p^3})=K_{\phi(p)}=K_{p-1}. $ Therefore

\begin{align*}
\Gamma(\mathbb{Z}_{p^4}) &=\Upsilon_{p^3}[\Gamma(A_{p}),\Gamma(A_{p^3}),\Gamma(A_{p^2})] =P_{3}[\overline{K}_{p^2(p-1)},K_{{p-1}},K_{p(p-1)}]\\
&=K_{p-1}\triangledown (\overline{K}_{p^2(p-1)}\cup K_{p(p-1)}) .\end{align*}

Thus, by Theorem <ref>, the distance signless Laplacian  spectrum of $ \Gamma(\mathbb{Z}_{p^4}) $ is

\begin{equation*}
\left\lbrace (n-2)^{[p-2]},(2n-p-3)^{[p^2(p-1)-1]},(2n-p^2-1)^{[p(p-1)-1]},x_{1}\geq x_{2}\geq x_{3} \right\rbrace,
\end{equation*}

where $ x_{1}, x_{2}, x_{3} $ are the zeros of the polynomial $ x^3- x^2(2 p^3-2p^2-p + 5 n-7)+x( p^5-p^4-6np^3+12p^3-17p^2-8np+17p-8n^2+38n-44) - p^7+ 4 p^6 - 7 p^5 -p^4(3 n-13) + p^3(4 n^2-10 n ) -p^2( 4 n^2-13n+9)+ p( 4 n^2- 16 n+16) + 4 n^3 - 28n^2+64n-48

§  SPECTRAL RADIUS OF GENERALIZED DISTANCE MATRIX OF   GRAPHS

The generalized distance matrix $\mathcal{D}_{\alpha}(G)$ of $ G $ introduced by Cui et al. [30] as a convex combinations of $Tr(G)$ and $\mathcal{D}(G)$, and is defined as $\mathcal{D}_{\alpha}(G)=\alpha Tr(G)+(1-\alpha)\mathcal{D}(G)$,  for $0\leq \alpha\leq 1$. Clearly,

\[\mathcal{D}_{0}(G)=\mathcal{D}(G), \qquad 2\mathcal{D}_{\frac{1}{2}}(G)=\mathcal{D}^{Q}(G), \qquad \mathcal{D}_{1}(G)=Tr(G)\]

\[\mathcal{D}_{\alpha}(G)-\mathcal{D}_{\beta}(G)=(\alpha-\beta)\mathcal{D}^{L}(G).\]

The matrix $ \mathcal{D}_{\alpha}(G) $ reduces to merging the distance spectral, distance Laplacian spectral and distance signless Laplacian spectral theories. Since the matrix $ \mathcal{D}_{\alpha}(G)$ is real symmetric, all its eigenvalues are real. Therefore, we can arrange them  as $ \partial_{1}\geq \partial_{2}\geq \dots \geq \partial_{n}$. The largest eigenvalue  $ \partial_{1} $ of the matrix $\mathcal{D}_{\alpha}(G)$ is called the generalized distance spectral radius of $ G$ ( we will denote $\partial_1(G)$ by $\partial(G)$). As $ \mathcal{D}_{\alpha}(G) $ is non-negative and irreducible, by the Perron-Frobenius theorem, $ \partial(G)$ is the unique eigenvalue and there is a unique positive unit eigenvector $X$ corresponding to $ \partial(G),$ which is called the generalized distance Perron vector of $G.$ Further results on generalized distance matrix can be seen in [31, 46, 126, 127, 128, 133].

For $ M\in \mathbb{M}_{mn}(\mathbb{R}),$ the Frobenius norm is defined as

\[ \parallel M \parallel_F =\sqrt{\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}|m_{ij}|^2}= \sqrt{trace(M^TM)} ,\]
where the trace of a square matrix is defined as sum of the diagonal entries. Further, if $ MM^T=M^TM $, then $ \parallel M \parallel_F^2=\sum\limits_{i=1}^{n}|\lambda_i(M)|^2 $, where $ \lambda_i's $ are the eigenvalues of $ M$. Also, $ \|M\|_{1}=|\lambda_{1}| $ is known as the spectral norm of $ M. $

We have the following observations.

Let $G$ be a connected graph of order $n$ having transmission degrees $Tr_1,Tr_2,\dots,Tr_n$ and Wiener index $W(G)$. Then

(1) $\sum\limits_{i=1}^{n}\partial_i=2\alpha W(G).$

(2) $\sum\limits_{i=1}^{n}\partial_i^2=\alpha^2 \sum\limits_{i=1}^{n}Tr_i^2+(1-\alpha)^2\parallel \mathcal{D}(G)\parallel_F^2$.

(3) $\sum\limits_{i=1}^{n}\left (\partial_i-\frac{2\alpha W(G)}{n}\right )^2=\alpha^2\sum\limits_{i=1}^{n}Tr_i^2+(1-\alpha)^2\parallel \mathcal{D}(G)\parallel_F^2-\dfrac{4\alpha^2W^2(G)}{n}$.

(4) $\partial(G)\geq \dfrac{2W(G)}{n}$, equality holds if and only if $G$ is transmission regular graph.

(5) $ \partial(G)\geq \sqrt{\dfrac{\sum\limits_{i=1}^{n}Tr_i^2}{n}} $, equality holds if and only if $G$ is transmission regular graph.

Proof. (1). We have $\sum\limits_{i=1}^{n}\partial_i=\alpha \sum\limits_{i=1}^{n}Tr_i+(1-\alpha)\sum\limits_{i=1}^{n}\rho_i^\mathcal{D}=\alpha\sum\limits_{i=1}^{n}Tr_i=2\alpha W(G)$.

(2). We have

\begin{align*}
&\sum\limits_{i=1}^{n}\partial_i^2 =\alpha^2\sum\limits_{i=1}^{n}Tr_i^2+(1-\alpha)^2\sum\limits_{i=1}^{n}(\rho_i^\mathcal{D})^2  =\alpha^2\sum\limits_{i=1}^{n}Tr_i^2+(1-\alpha)^2\sum\limits_{i=1}^{n}\sum\limits_{j=1,j\neq i}^{n}(d_{ij})^2\\
& =\alpha^2\sum\limits_{i=1}^{n}Tr_i^2+(1-\alpha)^22\sum\limits_{i=1}^{n-1}\sum\limits_{j=i+1}^{n}(d_{ij})^2 =\alpha^2\sum\limits_{i=1}^{n}Tr_i^2+(1-\alpha)^22|\sum\limits_{i=1}^{n-1}\sum\limits_{j=i+1}^{n}(d_{ij})^2|\\
& =\alpha^2\sum\limits_{i=1}^{n}Tr_i^2+(1-\alpha)^2\parallel \mathcal{D}(G)\parallel_F^2
\end{align*}

(3). This follows as a consequence of (2).

(4). From the Raleigh-Ritz's theorem for Hermitian matrices [92], we have

Then, for the unit vector U, we have
\[  \partial(G)\geq \dfrac{\textbf{U}^t\mathcal{D}_\alpha(G)\textbf{U}}{\textbf{U}^t\textbf{U}}=\dfrac{\alpha\sum\limits_{i=1}^{n}Tr_i+

Suppose $G$ is $k$ transmission regular. Then each row sum of $\mathcal{D}_\alpha(G)$ sums to a constant $k$. Therefore, by the Perron-Frobenius theorem [92], $k$ is simple and largest eigenvalue of $\mathcal{D}_\alpha(G)$, thus $\partial(G)=k$ and equality holds. Conversely, assume equality holds. Then $\partial(G )= \dfrac{\textbf{U}^t\mathcal{D}_\alpha(G)\textbf{U}}{\textbf{U}^t\textbf{U}}$. Hence $\mathcal{D}_\alpha(G)\textbf{U} =\partial(G)\textbf{U}$. Therefore $Tr_i=\partial(G)$ for all $i$ and thus $G$ is transmission regular.

(5). Proof is similar to that of (4).

Let $ H $ be a graph with  transmission degree sequence $ t_{i} $ such that there is a vertex $v_{i}$ having transmission degree equal to $t_{1}=Tr_{\max}$, transmission degree of every neighbor of $v_{i}$ equal to $t_{2}$ and transmission degree of every vertex non-adjacent to $v_{i}$ equal to $Tr_{\min}$.

We consider a column vector $X=[x_{1},x_{2},\ldots,x_{n}]^{T}\in \mathbb{R}^{n}$ to be a function defined on $ V(G) $ which maps vertex $ v_{i} $ to $ x_{i} $, that is,
$ X(v_{i})=x_{i} $ for $i$, $1\leq i \leq n$. The quadratic form $X^{T}\mathcal{D}_{\alpha}(G)X$ can be described as

\begin{eqnarray*}
X^{T}\mathcal{D}_{\alpha}(G)X= \alpha\sum_{i=1}^{n}t(v_{i})x_{i}^{2}+2(1-\alpha)\sum_{1\leq i<j\leq n}d(v_{i},v_{j})x_{i}x_{j},
\end{eqnarray*}

\begin{eqnarray*}
X^{T}\mathcal{D}_{\alpha}(G)X= (2\alpha-1)\sum_{i=1}^{n}t(v_{i})x_{i}^{2}+(1-\alpha)\sum_{1\leq i<j\leq n}d(v_{i},v_{j})(x_{i}+x_{j})^2.
\end{eqnarray*}

Also,  $\partial$ is an eigenvalue of $\mathcal{D}_{\alpha}(G)$ corresponding to the eigenvector $X$ if and only if
$X\neq \textbf{0}$ and

\begin{eqnarray*}
\partial x_v=\alpha Tr(v_{i})x_{i}+(1-\alpha)\sum_{j=1}^{n}d(v_{i},v_{j})x_{j}.
\end{eqnarray*}

These equations are called the $(\partial, x)$-eigenequations of $G$. For a normalized column vector $ X\in \mathbb{R}^{n} $ with at least one
non-negative component, by the Rayleigh's  principle, we have

$$\partial(G)\geq X^{T}\mathcal{D}_{\alpha}(G)X,$$

with equality if and only if $X$ is the generalized distance Perron vector of $G$.

The following result gives the upper bound for the largest eigenvalue of $ \mathcal{D}_{\alpha}(G). $

If $\partial(G)$ is the spectral radius of $\mathcal{D}_{\alpha}(G)$, then

\begin{align}\label{zero}
\partial(G) \leq \max\limits_{1\leq i,~j\leq n}& \frac{1}{2}\Big[\alpha t_{i}+t_{j}-(1-\alpha)d_{ij}\\\nonumber
&+\sqrt{(\alpha t_{i}-t_{j})^{2}+(1-\alpha)(1-\alpha-2t_{j}-4t_{i}-2\alpha t_{i})d_{ij}}\Big],
\end{align}

where $t_{max}=t_{1}\geq t_{2}\geq \dots \geq t_{n}=t_{min}$ are the vertex transmission degrees of $G$. Further, equality holds if and only if $G$ is a transmission regular graph or $G$ is a semi-regular graph $G(n-1,s)$ or $G$ is in $H$, where $H$ is defined above.

Proof.  Corresponding to the eigenvalue $\partial$ of $\mathcal{D}_{\alpha}(G)$, let $X=[x_{1},x_{2},\ldots,x_{n}]^{T}$ be the eigenvector, with $x_{i}=1$ and $x_{t}\geq 1$ for all $t\neq i$. Let $x_{j}=\max \{x_{t}: ~1\leq t\leq n ~and~ t\neq i \}$. Therefore, from the $i$-th and $j$-th equations of $\partial~X=\mathcal{D}_{\alpha}(G)~X$, we have

\begin{equation}\label{1}
\partial~x_{i}=\alpha t_{i}x_{i}+(1-\alpha)\sum\limits_{t=1, t\neq i}^{n} d_{it}x_{t}
\end{equation}

\begin{equation}\label{2a}
\partial~x_{j}=\alpha t_{j}x_{j}+(1-\alpha)\sum\limits_{t=1, t\neq j}^{n} d_{jt}x_{t}.
\end{equation}

(<ref>) implies that $\partial \leq \alpha t_{i}+(1-\alpha)t_{i}x_{j},$ so that

\begin{equation}\label{3a}
\partial -\alpha t_{i}\leq (1-\alpha)t_{i}x_{j}
\end{equation}

Also, (<ref>) implies that $\partial~x_{j}\leq \alpha t_{j}x_{j}+(1-\alpha)d_{ij}+ (1-\alpha)(t_{j}-d_{ij})x_{j}$,
which on simplification gives

\begin{equation}\label{4a}
\left[ \partial -t_{j}+(1-\alpha)d_{ij}\right]x_{j}\leq (1-\alpha)d_{ij}.
\end{equation}

Combining (<ref>) and (<ref>), we get

\begin{equation*}
(\partial-\alpha t_{i})[\partial-t_{j}+(1-\alpha)d_{ij}]x_{j}\leq [(1-\alpha)t_{i}x_{j}][(1-\alpha)d_{ij}].
\end{equation*}

Since $x_{j}>0$, therefore we have

\begin{equation*}
(\partial-\alpha t_{i})[\partial-t_{j}+(1-\alpha)d_{ij}]\leq (1-\alpha)^{2}t_{i}d_{ij},
\end{equation*}

which on simplification yields

\begin{equation}\label{5}
\partial^{2}-\left[\alpha t_{i}+t_{j}-(1-\alpha)d_{ij}\right]\partial+\alpha t_{i}t_{j}-(1-\alpha) t_{i}d_{ij}\leq 0.
\end{equation}

The solution of (<ref>) is given by

\begin{equation*}
\partial \leq \frac{1}{2}\left[\alpha t_{i}+t_{j}-(1-\alpha)d_{ij}+\sqrt{[\alpha t_{i}+t_{j}-(1-\alpha)d_{ij}]^{2}-4t_{i}\left[\alpha t_{j}+(1-\alpha)d_{ij}\right]}\right]
\end{equation*}

\begin{equation*}
\partial \geq \frac{1}{2}\left[\alpha t_{i}+t_{j}-(1-\alpha)d_{ij}-\sqrt{[\alpha t_{i}+t_{j}-(1-\alpha)d_{ij}]^{2}-4t_{i}\left[\alpha t_{j}+(1-\alpha)d_{ij}\right]}\right].
\end{equation*}

Therefore, in all cases, it follows that
\begin{align*}
\partial(G)\leq \max\limits_{1\leq i,~j\leq n} &\frac{1}{2}\Big[\alpha t_{i}+t_{j}-(1-\alpha)d_{ij}\\
&+\sqrt{[\alpha t_{i}+t_{j}-(1-\alpha)d_{ij}]^{2}-4t_{i}[\alpha t_{j}+(1-\alpha)d_{ij}]}\Big].
\end{align*}

Now, we characterize the graphs for which equality holds in (<ref>). In this regard, first assume that equality holds in (<ref>). Therefore, equality cases in (<ref>) and (<ref>) imply that $x_{t}=x_{j}$ for all $t$, with $t\neq i$. As $x_{j}\leq x_{i}=1$, we have the following two possibilities to consider.

Case 1. $x_{j}= x_{i}=1$. Here, $\partial =t_{i}$ for all $i$, $1\leq i\leq n$. So $G$ is a transmission regular graph.

Case 2. $x_{j}< x_{i}=1$. Therefore, either $d_{i}=n-1$ or $d_{i}<n-1$. We look at these two cases separately as follows.

Case 2.1. If $d_{i}=n-1$, then clearly $v_{t}\in N(v_{i})$ for all those $v_{t}$ which are in $V(G)-\{v_{i}\}$. Now, for any $v_{t}\in V(G)-\{v_{i}\}$, we have $\partial x_{j}=(t_{t}+\alpha -1) x_{j} +(1-\alpha)$. This clearly indicates that transmission degree of every vertex $v_{t}\in V(G)-\{v_{i}\}$ is equal to $t_{t}$, with of course $t_{t}> n-1$. So, evidently $t_{2}=t_{3}=\dots =t_{n}=t_{min}$. Further, $d_{i}=n-1$ implies that $G$ is of diameter 2 and thus $t_{i}=2n-2-d_{i}$. Therefore, the vertex degrees of $G$ are $n-1$, $d_{2}=d_{3}=\dots =d_{n}=s$ (say) with clearly $n-1>s$. Hence $G$ is isomorphic to $G(n-1,s)$.

Case 2.2.  Now, for the case $d_{i}<n-1$, we consider the vertex partition of $V(G)-\{v_{i}\}$ as $U=N(v_{i})$ and $W=V(G)-(U\cup \{v_{i}\})$. Then

\begin{equation}\label{11a}
\partial =t_{i}\left[\alpha +(1-\alpha)x_{j}\right] ~~for~~v_{i},
\end{equation}

\begin{equation}\label{12}
\partial x_{j}=\left[t_{t}-(1-\alpha)\right]x_{j}+(1-\alpha) ~~for~~v_{t}\in N(v_{i}),
\end{equation}

\begin{equation}\label{13a}
\partial =t_{k}~~~~for~~~~v_{k}\notin \left[N(v_{i})\cup \{v_{i}\}\right].
\end{equation}

Simplifying (<ref>) implies that $t_{t}=\partial +(1-\alpha)(x_{j}-1)\frac{1}{x_{j}}$, for all $v_{t}\in N(v_{i})$. Therefore, transmission degree of every vertex in $N(v_{i})$ is equal to $t_{t}$, and transmission degree of every vertex outside $N(v_{i})\cup \{v_{i}\}$ is equal to $t_{i}$. Since $\alpha +(1-\alpha)x_{j}\leq 1$, from (<ref>) and (<ref>), we have $t_{k}>t_{i}$. In a similar way, $t_{k}>t_{t}$ follows from (<ref>) and (<ref>). Thus, $t_{i}=t_{max}$, and $t_{k}=t_{2}$ for all $v_{k}\in W$. Also, $t_{t}=t_{min}$ for all vertices in $U$.

From the arguments given above, we conclude that the given connected graph contains a vertex $v_{i}$ with transmission degree $t_{1}=t_{max}$, transmission degree of every neighbor of $v_{i}$ is equal to $t_{2}$ and transmission degree of every vertex non-adjacent to $v_{i}$ is equal to $t_{min}$. This is clearly the graph $H$ defined above.

Conversely, for the regular graph, or the graph isomorphic to $G(n-1,s)$, or graph $H$, it is easy to verify that equality holds.

§ GENRALIZED DISTANCE ENERGY

Let the auxiliary eigenvalues $ \Theta_i $, corresponding to the generalized distance eigenvalues be

\begin{equation*}
\Theta_i=\partial_i-\frac{2\alpha W(G)}{n}.
\end{equation*}

Now, we define the generalized distance energy of $G$ as the mean deviation of the  values of the generalized distance eigenvalues of $G$, that is,

\begin{equation*}
E^{\mathcal{D}_{\alpha}}(G)=\sum_{i=1}^{n}\left|\partial_i-\frac{2\alpha W(G)}{n}\right|=\displaystyle\sum_{i=1}^{n}|\Theta_{i}|.
\end{equation*}

We note that $ \sum\limits_{i=1}^{n}
|\Theta_{i}| $ is the trace norm of the matrix $ \mathcal{D}_{\alpha}(G)-\frac{2W(G)}{n}I_{n} $, where $ I_{n} $ is the identity matrix. It is easy to see that $\sum_{i=1}^{n}\Theta_{i}=0.$  Clearly, $E^{\mathcal{D}_{0}}(G)=E^{\mathcal{D}}(G)$ and $2E^{\mathcal{D}_{\frac{1}{2}}}(G)=E^{Q}(G).$ This shows that the concept of generalized distance energy of $G$ merges the theories of distance energy and the distance signless Laplacian energy of $G$. Therefore, it will be interesting to study the quantity $E^{\mathcal{D}_{\alpha}}(G)$ and explore some properties like the bounds, the dependence on the structure of graph $G$ and the dependence on the parameter $\alpha$.

The following lemma can be found in [30].

Let $G$ be a connected graph of order $n$ and $ \frac{1}{2}\leq \alpha\leq 1 .$ If $ G'$ is a connected graph obtained from $ G $ by deleting an edge, then  $ \partial_i(\mathcal{D}_\alpha(G'))\geq\partial_i(\mathcal{D}_\alpha(G))$, for all $ 1\leq i\leq n$.

The following result shows that the complete bipartite graph has the minimum generalized distance energy among all connected bipartite graphs of order $n$.

Let $G$ be a connected  bipartite graph of order $n\geq 2$ having partite sets of cardinality $a$ and $n-a$, with $n-a\geq a,a\geq 2$ and let $\alpha\in [\frac{1}{2},1)$.

(1) If $2a<n< \frac{2a^2}{a-2}$ and $\alpha< \gamma$ or $\frac{2a^2}{a-2}\le n< 4.877 a$, then for $t=n-a$, we have

\[ E^{\mathcal{D}_\alpha}(G)\geq (\alpha+2)n-4+\sqrt{\Delta}+2(n-a-1)(\alpha(2n-a)-2)-\frac{4\alpha(n-a) W(G)}{n}, \]

and for $t=n-a+1$, we have

\[ E^{\mathcal{D}_\alpha}(G)\geq 2(\alpha+2)n-8+2(n-a-1)(\alpha(2n-a)-2)-\frac{4\alpha(n-a+1) W(G)}{n}. \]

(2) If $n\ge 4.877 a$, then
\[ E^{\mathcal{D}_\alpha}(G)\geq (\alpha+2)n-4+\sqrt{\Delta}+2(n-a-1)(\alpha(2n-a)-2)-\frac{4\alpha(n-a) W(G)}{n}. \]

(3) If $n=2a$, then
\[ E^{\mathcal{D}_\alpha}(G)\geq 2(2a\alpha+4a-4)-\frac{4\alpha W(G)}{n}, \]
where $\Delta=(2a^2+n^2-2na)(\alpha-2)^2+2(na-a^2)(\alpha^2-2)$, $\gamma=\frac{2n}{2n+an-2a^2}$ and $t$ is the number of generalized distance eigenvalues greater than or equal to $\frac{2\alpha W(K_{a,n-a})}{n}$.
Equality occurs in each case if and only if $G\cong K_{a,n-a}$.

Let $ G $ be a connected bipartite graph of order $n$  with vertex set $V(G)$. Let $V(G)=V_1\cup V_2$, with $|V_1|=a, |V_2|=n-a$ and $n-a\geq a,~ a\geq 2$, be the bipartition of the vertex set  of $G$. Clearly $ G $ is a spanning subgraph of the complete bipartite graph $ K_{a,n-a}.$ Therefore, by Lemma <ref> with $ \frac{1}{2}\leq \alpha\leq 1 $, we have $ \partial_i(G)\geq\partial_i(K_{a,n-a})$, for all $i=1,2,\dots,n$.
Let $ s$ be a positive integer such that $ \partial_s\geq\frac{2\alpha W(G)}{n}$ and $ \partial_{s+1}<\frac{2\alpha W(G)}{n} $. Using $\sum\limits_{i=1}^{n}\partial_i=2\alpha W(G)$ and the definition of generalized distance energy, we have

\begin{align*}
E^{\mathcal{D}_\alpha}(G)&=\sum\limits_{i=1}^{n}\left|\partial_i-\frac{2\alpha W(G)}{n}\right|\\&=\sum\limits_{i=1}^{s}\left( \partial_i-\frac{2\alpha W(G)}{n}\right) +\sum\limits_{i=s+1}^{n}\left( \frac{2\alpha W(G)}{n}-\partial_i\right)\\
&=2\left(\sum\limits_{i=1}^{s}\partial_i-\frac{2s\alpha W(G)}{n} \right).
\end{align*}

We first show that

\begin{equation}\label{def}
E^{\mathcal{D}_\alpha}(G)=2\left(\sum\limits_{i=1}^{s}\partial_i-\frac{2s\alpha W(G)}{n} \right)=2 \max_{1\leq j\leq n}\left\lbrace \sum\limits_{i=1}^{j}\partial_i-\frac{2\alpha jW(G) }{n}\right\rbrace.
\end{equation}

If $ j>s$, we have

\begin{align*}
\sum\limits_{i=1}^{j}\partial_i -\frac{2\alpha jW(G)}{n} &
=\sum\limits_{i=1}^{s}\partial_i+\sum\limits_{i=s+1}^{j}\partial_i-\frac{2\alpha jW(G)}{n} \\
& < \sum\limits_{i=1}^{s}\partial_i-\frac{2\alpha sW(G)}{n}
\end{align*}

as $\partial_i<\frac{2\alpha W(G)}{n}$, for $i\geq s+1$. Similarly, for $ j\leq s$, we have

\begin{align*}
\sum\limits_{i=1}^{j}\partial_i -\frac{2\alpha jW(G)}{n}\leq \sum\limits_{i=1}^{s}\partial_i-\frac{2\alpha sW(G)}{n}.
\end{align*}

This proves (<ref>). Let $t$ be the number of generalized distance eigenvalues of $K_{a,n-a}$ which are greater than or equal to $\frac{2\alpha W(K_{a,n-a})}{n}$ Then $1\le t\le n$. Therefore, for $n\geq 2a$ and $\alpha\in [\frac{1}{2},1)$, from (<ref>), we have

\begin{align}\label{y1}
E^\mathcal{D}_{\alpha}(G)=&2\max_{1\leq j\leq n}\left\lbrace \sum\limits_{i=1}^{j}\partial_i-\frac{2\alpha jW(G) }{n}\right\rbrace\geq 2\sum\limits_{i=1}^{t}\partial_i(G)-\frac{4t\alpha W(G) }{n}\nonumber\\&
\geq   2\sum\limits_{i=1}^{t}\partial_i(K_{a,n-a})-\frac{4t\alpha W(G)}{n},
\end{align}
where $t$ is a positive integer defined above.

In [126], it is shown that the generalized distance spectrum of $ K_{a,n-a} $ is

$$\left\lbrace  (\alpha(a +n)-2)^{[a-1]},(\alpha(2n-a)-2)^{[n-a-1]},\frac{\alpha n+2n-4 \pm \sqrt{\Delta}}{2} \right\rbrace ,$$

where $ \Delta = (2a^2+n^2-2na)(\alpha-2)^2+2(na-a^2)(\alpha^2-2)$ and $ \partial_1=\frac{\alpha n+2n-4 + \sqrt{\Delta}}{2} $ is the spectral radius. Also, we have $\frac{2\alpha W(K_{a,n-a})}{n}=\frac{\alpha(2a^2+2n^2-2n-2na) }{n}.$ When $\frac{1}{2}\leq\alpha< 1$, we will show that $t=1$ or  $2$ or $n-a$ or $n-a+1$. It is easy to see that $\alpha(2n-a)-2\geq \alpha(n+a)-2$, for all $n-a\geq a$. We claim that $\alpha(2n-a)-2$ is the second largest generalized distance eigenvalue of $K_{a,n-a}$ for $n\ge 3.5 a$. We have

\begin{align*}
\alpha(2n-a)-2\ge \frac{(\alpha+2)n-4- \sqrt{\Delta}}{2},
\end{align*}
which implies that

\begin{align}\label{xa}
\sqrt{\Delta}\ge 2n-(3n-2a)\alpha.
\end{align}
Since right hand side of (<ref>) is less or equal to zero  for $\frac{2n}{3n-2a}\leq \alpha\leq 1$, it follows that our claim is true for $\alpha\in \left [\frac{2n}{3n-2a},1\right ]$. So, suppose that $\alpha\in \left [\frac{1}{2},\frac{2n}{3n-2a}\right )$, and for this value of $\alpha$, from (<ref>), we have $4(-2n^2+3an-a^2)\alpha^2+8\alpha(n^2-a^2)-12a(n-a)\geq 0,$ that is,

\begin{align*}
-4(n-a)\Big((2n-a)\alpha^2-2\alpha(n+a)+3a\Big)\geq 0.
\end{align*}

Consider the function $f(\alpha)=(2n-a)\alpha^2-2\alpha(n+a)+3a$, for all $\alpha\in \left [\frac{1}{2},\frac{2n}{3n-2a}\right )$. It is easy to see that this function is increasing for $\alpha\geq \delta$ and decreasing for $\alpha<\delta$, where $\delta=\frac{n+a}{2n-a}$. Clearly,  $\delta\in \left [\frac{1}{2},\frac{2n}{3n-2a}\right )$, for $n>2a$. Since $f(\alpha)$ is increasing for $\alpha\geq \delta$ and decreasing for $\alpha<\delta$, it follows that the sign of $f(\frac{1}{2})$ will decide the nature of $f(\alpha)$ in $\left [\frac{1}{2},\delta\right ]$ and sign of $f(\frac{2n}{3n-2a})$ will decide the nature for $f(\alpha)$ in $[\delta,\frac{2n}{3n-2a}]$. We have $f(\frac{1}{2})=-\frac{1}{4}(2n-7a)\le 0$, for all $n\ge \frac{7a}{2}$. This shows that Inequality  (<ref>) holds for all $n\ge 3.5 a$.  If $2a< n<3.5 a$, then for some values of $\alpha\in [\frac{1}{2},\delta)$ the eigenvalue $\alpha(2n-a)-2$ is the second largest, while as for some other values $\alpha\in [\frac{1}{2},\delta)$ the eigenvalue $\frac{(\alpha+2)n-4- \sqrt{\Delta}}{2}$ is the second largest. For $2a< n\le 3a$, it is easy to see that $\delta\ge 0.8$ implying that $0.8\in [\frac{1}{2},\delta)$. We have $f(0.8)=\frac{1}{25}(19a-8n)\le 0$, provided that $n\ge 2.375 a$. This gives that $f(0.8)> 0$ for  $2a< n< 2.375a$ and  $f(0.8)\le 0$ for  $2.375a\le n\le 3a$. Again $0.6, 0.7\in  [\frac{1}{2},\delta)$, for all $2a< n<3.5 a$ and we have $f(0.6)=\frac{1}{25}(36a-12n)$ and $f(0.7)=\frac{1}{100}(111a-42n)$, giving that $f(0.6)> 0$ for  $2a\le n< 3a$ and  $f(0.6)\le 0$ for  $3a\le n< 3.5a$, while as $f(0.7)> 0$ for  $2a\le n< \frac{37a}{14}$ and  $f(0.7)\le 0$ for  $\frac{37a}{14}\le n< 3.5a$.  Further, for $\alpha\in [\delta,\frac{2n}{3n-2a}]$, we have $f(\frac{2n}{3n-2a})=(2n-a)(\frac{2n}{3n-2a})^2-2(n+a)(\frac{2n}{3n-2a})+3a=\frac{1}{(3n-2a)^2}(n-a)(n-2a)^2(3a-4n)<0$, for all $n\ge 2a$.  Thus, it follows that for $\alpha\in [0.5,\delta)$ with $n\ge 3.5 a$ and for  $\alpha\in [\delta,1)$ with $n> 2a$, the eigenvalue  $\alpha(2n-a)-2$ is the second largest. For $\alpha\in [0.5,\delta)$ with $2a< n<3.5 a$, there are two choices for the second largest eigenvalue, depending upon the value of $\alpha$. In fact, if $\alpha\in [0.5,0.6]$, then for $2a<n\le 3a$, the eigenvalue $\frac{(\alpha+2)n-4- \sqrt{\Delta}}{2}$  is the second largest, while as the eigenvalue $\alpha(2n-a)-2$ is the third largest and for $3a<n< 3.5a$, the eigenvalue $\alpha(2n-a)-2$  is the second largest, while as the eigenvalue $\frac{(\alpha+2)n-4- \sqrt{\Delta}}{2}$  is the third largest. For $\alpha\in [0.6, \delta)$, we have both the possibilities. For the eigenvalue $\frac{(\alpha+2)n-4- \sqrt{\Delta}}{2}$ and $ \alpha(n+a)-2$, proceeding similarly it can be seen that $\frac{(\alpha+2)n-4- \sqrt{\Delta}}{2}\geq \alpha(n+a)-2$ for all $\alpha\in [\frac{1}{2},1]$ and $a\ne 1$. Thus it follows that for $\alpha\in [0.5,\delta)$ with $n\ge 3.5 a$ and for  $\alpha\in [\delta,1)$ with $n>2a$, $a\ne 1$, we have

\begin{align*}
\frac{\alpha n+2n-4 +\sqrt{\Delta}}{2}\geq (\alpha(2n-a)-2)\geq\frac{\alpha n+2n-4 -\sqrt{\Delta}}{2}\geq (\alpha(a +n)-2).
\end{align*}
For  $\alpha\in [0.5,\delta)$ with $2a\le n<3.5 a$, we have

\begin{align*}
\frac{\alpha n+2n-4 +\sqrt{\Delta}}{2}\geq \phi_1\geq\phi_2\geq (\alpha(a +n)-2),
\end{align*}
where $\phi_1=\max\{(\alpha(2n-a)-2),\frac{\alpha n+2n-4 -\sqrt{\Delta}}{2}\}$ and  $\phi_2=\min\{(\alpha(2n-a)-2),\frac{\alpha n+2n-4 -\sqrt{\Delta}}{2}\}$.
For the eigenvalue $\alpha(n+a)-2$, it is easy to see that $\alpha(n+a)-2< \frac{2\alpha W(K_{a,n-a})}{n}$ holds for all $n> 2a$ and $\alpha\ge 0.5 $. For the eigenvalues $(\alpha(2n-a)-2)$, with $n>2a$, we have

\begin{align*}
\frac{2\alpha W(K_{a,n-a})}{n}=\frac{\alpha(2a^2+2n^2-2n-2na) }{n}\leq (\alpha(2n-a)-2),
\end{align*}

which gives

\begin{align}\label{x1}
\alpha(2a^2-2n-an)+2n\leq 0.
\end{align}

This in turn implies that  (<ref>) is true for  $\alpha\geq \gamma$, where $\gamma=\frac{2n}{2n+an-2a^2}$. For $a\geq 2$, it can be seen that $\gamma<\frac{1}{2}$, for all $n\ge \frac{2a^2}{a-2}$ implying that   (<ref>) is true for all $\alpha\geq \frac{1}{2}$. Since $3.5 a\ge \frac{2a^2}{a-2}$, for all $a\ge 5$, it follows that Inequality (<ref>) is true for all $n\ge 3.5 a$ with $a\ge 5$ and is true for $2\le a\le 4$, provided that $n\ge \frac{2a^2}{a-2}$. So, suppose $3.5a\leq n<\frac{2a^2}{a-2}$. For this value of $n$, it is clear that $\gamma\in [\frac{1}{2},1]$ and therefore Inequality (<ref>) holds for $ \alpha\geq \gamma$ and does not hold for $\alpha<\gamma$. This shows that for $a\geq 5$ and  $n\ge 3.5 a$; for $2\le a\le 4$ and $n\ge \frac{2a^2}{a-2}$; for  $3.5a\leq n<\frac{2a^2}{a-2}$ and $\alpha\geq \gamma$, we have $t\geq n-a$; while for $3.5a\leq n<\frac{2a^2}{a-2}$ and $\alpha<\gamma$, we have $t=1$ or $2$.

For  $n\ge 3.5a$, we have

\begin{align*}
\frac{\alpha n+2n-4 - \sqrt{\Delta}}{2}< \frac{\alpha(2a^2+2n^2-2n-2na) }{n},
\end{align*}

\begin{align}\label{x2}
(2-3\alpha)n^2+4(\alpha-1)n-4a\alpha(a-n)< n\sqrt{\Delta}.
\end{align}

Consider the function $h(\alpha)=(2a^2+n^2-2na)(\alpha-2)^2+2(na-a^2)(\alpha^2-2)$, for $\alpha\in [\frac{1}{2},1]$. It is easy to see that the function $h(\alpha)$ is a decreasing function of $\alpha$. Therefore the minimum value of $h(\alpha)$ is attained at $\alpha=1$, that is, $h(\alpha)\geq h(1)=(n-2a)^2$. With this, from (<ref>), we have
$(2-3\alpha)n^2+4(\alpha-1)n-4a\alpha(a-n)< n(n-2a)$, implies that $n^2+2an-4n<\alpha(3n^2-4n(a+1)+4a^2)$,
which in turn implies that $\alpha >\frac{n^2+2an-4n}{3n^2-4n(a+1)+4a^2}=\gamma_1$. This implies that inequality <ref> holds for $\alpha\in (\gamma_1,1)$. It is easy to see that for $n\ge 3.5 a$, we have $\gamma_1< \frac{3}{4}$. Since $h(\alpha)$ is a decreasing function of $\alpha$, therefore $h(\alpha)\ge h(\gamma_1)>h(\frac{3}{4})$, for all $\alpha\in [0.5,\gamma_1]$. We have $ h(\frac{3}{4})=\frac{1}{16}\Big(25n^2-96an+96a^2\Big)> \frac{1}{16}\Big(5n-10a\Big)^2$. With this it follows from inequality <ref> that for $\alpha\in [0.5,\gamma_1]$, we have $(2-3\alpha)n^2+4(\alpha-1)n-4a\alpha(a-n)< n\Big(\frac{5n-10a}{4}\Big)$ implying that $\alpha> \frac{3n^2+10an-16n}{4(3n^2-4n(a+1)+4a^2)}=\gamma_2$. This shows that Inequality (<ref>) holds for $\alpha\in (\gamma_2,1)$. It is easy to see that for $n\ge 3.5 a$, we have $\gamma_2< \frac{2}{3}$. Again using the fact that $h(\alpha)$ is a decreasing function of $\alpha$, it follows that $h(\alpha)\ge h(\gamma_2)>h(\frac{2}{3})$, for all $\alpha\in [0.5,\gamma_2]$.  We have $ h(\frac{2}{3})=\frac{1}{9}\Big(16n^2-60an+60a^2\Big)$. From Inequality (<ref>), it follows that for $\alpha\in [0.5,\gamma_2]$, we have $(2-3\alpha)n^2+4(\alpha-1)n-4a\alpha(a-n)< \frac{n}{3}\sqrt{16n^2-60an+60a^2}$ implying that $\alpha> \frac{6n^2-12n-n\sqrt{16n^2-60an+60a^2}}{3(3n^2-4n(a+1)+4a^2)}=\gamma_3$. This shows  that Inequality (<ref>) holds for $\alpha\in (\gamma_3,1)$. It is easy to see that $\gamma_3\le 0.5$, for $n\ge 4.877 a$, implying that the Inequality (<ref>) holds for $\alpha$, provided that $n\ge 4.877 a$. For $2a< n< 4.877 a$, the Inequality (<ref>) does not hold for some values of $\alpha$ as can been seen as follows. Since $h(\alpha)$ is a decreasing function of $\alpha$, it follows that $h(\alpha)\le h(\frac{1}{2})=\frac{1}{4}\Big(9n^2-32an+32a^2\Big)$. Therefore, from Inequality (<ref>), it follows that $(2-3\alpha)n^2+4(\alpha-1)n-4a\alpha(a-n)< \frac{n}{2}\sqrt{9n^2-32an+32a^2}$ implying that $\alpha> \frac{2n^2-4n-0.5 n\sqrt{9n^2-32an+32a^2}}{3n^2-4n(a+1)+4a^2}=\gamma_4$. If $\gamma_4\le 0.5$, then

\begin{align}\label{ya}
8n^4-(40a-8)n^3+(24a^2+32a-16)n^2+32na^2(a-1)-16a^4\ge 0.
\end{align}

Consider the function $p(n)=nq(n)-16a^4$, where $q(n)=8n^3-(40a-8)n^2+(24a^2+32a-16)n+32a^2(a-1)$. It is easy to see that the function $q(n)$ is  decreasing in $(2a,\psi)$ and increasing in $[\psi, 4.877a)$. Since $\psi<3a$, so we have $q(2a)=-16a(a^2-4a+2)<0$, for $a\ge 2$, and $q(4a-3)=-8(a-3)(5a-5)\le 0$, for $a\ge 2$. For $2a< n< 4a-3$, we have $q(n)<0$ implying that $p(n)<0$ in this range of $n$. From this, it follows that for $2a< n< 4a-3$ with $\alpha \in [0.5,\gamma_4]$, the Inequality  (<ref>) does not hold.

Thus, it follows that if $n\ge 4.877 a$, then $t=n-a$; if $\frac{2a^2}{a-2}\le n< 4.877 a$, then $t=n-a$ or $t=n-a+1$; if $2a< n< \frac{2a^2}{a-2}$ and $\alpha\ge \gamma$, then $t=n-a$ or $t=n-a+1$ and if $2a< n< \frac{2a^2}{a-2}$ and $\alpha< \gamma$, then $t=1$ or $t=2$. Using this information, part (1) and (2) now follows by direct calculation from (<ref>).

If $n=2a$, then the generalized distance spectrum of $ K_{a,n-a} $ is $\{3a-2, 2a\alpha+a-2,3a\alpha-2^{[2a-2]}\}$, with $3a-2\ge 2a\alpha+a-2\ge 3a\alpha-2$. Also, $\frac{2\alpha W(K_{a,a})}{n}=\alpha(3a-2)$. It is easy to see that $t=2$, in this case. With this the result now follows by direct calculation from (<ref>). This completes the proof.

The following lemma can be found in [31].

Let $ T $ be a tree of order $ n\geq 4.$ If $ 0\leq\alpha<1,$ then
$$\partial(T)\geq \partial(S_{n})= \frac{\alpha n+2n-4+\sqrt{k}}{2},$$

where $ k=(n^2-2n+2)(\alpha-2)^2+2(n-1)(\alpha^2-2), $ with equality if and only if $ T\cong S_{n}. $

The following result gives the generalized distance energy of the star graph of order $n$.

For $\alpha\in (0,1)$, the generalized distance energy of the star graph $ S_{n}$ of order $n, n\geq 4$, is

\begin{equation*}
E^\mathcal{D}_{\alpha}(S_{n}) = \begin{cases}
\sqrt{k}+2n+\alpha(8-3n)-4-\frac{4\alpha}{n}, &0<\alpha<\frac{2n}{3n-2}\\
\sqrt{k}-n\alpha(4n-19)-22\alpha -2n+4+\frac{8\alpha}{n}, &\frac{2n}{3n-2}\leq \alpha<1
\end{cases},
\end{equation*}

where $ k=(n^2-2n+2)(\alpha-2)^2+2(n-1)(\alpha^2-2).$

Let $t$ be the greatest positive integer such that $\partial_t(S_{n})\geq \frac{2\alpha W(S_{n})}{n}$. The generalized distance spectrum of $ S_{n} $ is
$$\left\lbrace  (\alpha(2n-1)-2)^{[n-2]}, \frac{\alpha n+2n-4\pm \sqrt{k}}{2}\right\rbrace,$$
where $k=(n^2-2n+2)(\alpha-2)^2+2(n-1)(\alpha^2-2)$. For $n\geq 4$, from Lemma <ref>, it follows that
$ \frac{\alpha n+2n-4+ \sqrt{k}}{2}$ is the generalized distance spectral radius of $ S_{n}$, which is always greater or equal to $\frac{2\alpha W(S_{n})}{n}=\alpha(2n-4+\frac{2}{n})$. It is easy to verify that $ \frac{\alpha n+2n-4- \sqrt{k}}{2} $ is the least eigenvalue of $ S_{n}.$ For the eigenvalue $ \alpha(2n-1)-2$, we have two different cases.

Case (i). If $ 0<\alpha<\frac{2n}{3n-2}$, then it is easy to see that $ \alpha(2n-1)-2<\alpha(2n-4+\frac{2}{n})$ and so $t=1$. Therefore, using (<ref>), we have

\begin{align*}
E^{\mathcal{D}_\alpha}(S_{n})&=2\left(\sum\limits_{i=1}^{t}\partial_i-\frac{2t\alpha W(G)}{n} \right)\\&
= \sqrt{k}+2n+\alpha(8-3n)-4-\frac{4\alpha}{n}.
\end{align*}

Case (ii). If $ \frac{2n}{3n-2}\leq \alpha <1$, then it is easy to see that $ \alpha(2n-1)-2\geq\alpha(2n-4+\frac{2}{n})$ and so $t=n-1$. Therefore, using (<ref>), we have

\begin{align*}
E^{D_\alpha}(S_{n})&=2\left(\sum\limits_{i=1}^{t}\partial_i-\frac{2t\alpha W(G)}{n} \right)\\&
=2\sum\limits_{i=1}^{n-1}\partial_i- 4\alpha(n-1)(2n-4+\frac{2}{n})\\&
=\sqrt{k}+2n\alpha(n-2)+2\alpha-\alpha n+4-4\alpha(n-1)(2n-4+\frac{2}{n})\\&
= \sqrt{k}-n\alpha(4n-19)-22\alpha -2n+4+\frac{8\alpha}{n}.
\end{align*}

Now we obtain a lower bound for the generalized distance energy of a tree in terms of the order $n$, the Wiener index  and the parameter $\alpha$.

Let $T$ be a tree of order $ n$ and let $\alpha\in (0,1)$. Then

\begin{equation}\label{sl}
E^{D_\alpha}(T)\geq\alpha n+2n-4+\sqrt{k}- \frac{4\alpha W(G)}{n},
\end{equation}
where $ k=(n^2-2n+2)(\alpha-2)^2+2(n-1)(\alpha^2-2)$. Equality occurs for $ 0<\alpha<\frac{2n}{3n-2}$, if and only if $T\cong S_{n}$.

Let $T$ be a tree of order $n$ and let $t$ be the largest positive integer such that $\partial_t(T)\geq \dfrac{2\alpha W(T)}{n}$.  Using  (<ref>) and Lemma <ref>, we have

\begin{align*}
E^{\mathcal{D}_\alpha}(T)&=2\left(\sum\limits_{i=1}^{t}\partial_i(T)-\frac{2t\alpha W(T)}{n} \right)=2\max_{1\leq j\leq n}\left\lbrace \sum\limits_{i=1}^{j}\partial_i(T)-\frac{2\alpha jW(T) }{n}\right\rbrace \\&
\geq 2(\partial_1(T)-\dfrac{2\alpha W(T)}{n})
\geq 2\partial_1(S_{n})- \frac{4\alpha W(T)}{n}\\
& \geq \alpha n+2n-4+\sqrt{k}- \frac{4\alpha W(T)}{n},
\end{align*}
where $ k=(n^2-2n+2)(\alpha-2)^2+2(n-1)(\alpha^2-2)$. Assume that equality occurs  in (<ref>), then all the inequalities above occur as equalities. Since equality occurs in Lemma <ref> for $T\cong S_{n}$, from (<ref>), it follows  that  equality occurs in (<ref>) if and only if $t=1$ and $T\cong S_{n}$. Now using Lemma <ref> the result follows.

For $ 0<\alpha<\frac{2n}{3n-2}$, Theorem <ref> shows that among all trees of order $n$ the star graph $S_{n}$ has the minimum generalized distance energy.

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Papers Accepted/ Published:

* S. Pirzada, Hilal A. Ganie, Bilal A. Rather and R. U. Shaban, On the generalized distance energy of graphs, Linear Algebra and its Applications 603 (2020) 1–19.

SCI, Scopus, ISSN: 0024-3795.

* Hilal A. Ganie, S. Pirzada, Bilal A. Rather and V. Trevisan, Further developments on Brouwer's conjecture for the sum of Laplacian eigenvalues of graphs, Linear Algebra  and its Applications 588 (2020) 1–18.

SCI, Scopus, ISSN: 0024-3795.

* S. Pirzada, Bilal A. Rather, M. Aijaz and T. A. Chishti, Distance signless Laplacian spectrum of graphs and spectrum of zero divisor graphs of  $ \mathbb{Z}_{n} $,  Linear and MultiLinear Algebra (2020) DOI:10.1080/03081087.2020.1838425.

SCI, Scopus, ISSN: 03081087

* S. Pirzada, Bilal A. Rather, Hilal A. Ganie and R. U. Shaban, On Brouwers conjecture and spectrally threshold graphs,
accepted for publication in Journal of the Ramanujan Mathematical Society (January 2021).

SCI, Scopus, ISSN: 23203110 (online), ISSN: 09701249 (print).

* S. Pirzada, Bilal A. Rather, Hilal A. Ganie, and R. U. Shaban, On generalized distance spectral radius of a bipartite graph, Matematic̆ki Vesnik 72(4) (2020) 327–336.

Web of Science (ESCI), Scopus, ISSN: 2406-0682 (online), ISSN: 0025-5165 (print).

* S. Pirzada, Bilal A. Rather, R. U. Shaban and T. A. Chishti, On the sum of powers of $ A_{\alpha} $ eigenvalues of graphs and $ A_{\alpha} $-energy like invariant, Boletim da Sociedade Paranaese de Matemática (2020) DOI:10.5269/bspm.52469.

Web of Science (ESCI), Scopus, ISSN: 2175-1188 (online), ISSN: 0037-8712 (print).

* S. Pirzada, Bilal A. Rather and R. U. Shaban, On graphs with minimal distance signless Laplacian energy, accepted for publication in  Acta Universitatis Sapientiae Mathematica ( August 2020).

Web of Science (ESCI), Scopus, ISSN: 2066-7752 (online), ISSN: 1844-6094 (print).

* T. A. Naikoo, U. Samee, S. Pirzada and Bilal A. Rather, On degree sets in k-partite graphs, Acta Universitatis Sapientae Informatica 12(2) (2020) 251–259.

Web of Science (ESCI), ISSN: 2066-7760 (online), ISSN: 1844-6086 (print).

* S. Pirzada, Bilal A. Rather and R. U. Shaban and Merajuddin, On signless Laplacian spectrum of zero divisor graphs of the ring $ \mathbb{Z}_{n} $, accepted for publication in   The Korean Journal of Mathematics ( Jan 2021).

Web of Science (ESCI), ISSN: 2288-1433 (online), ISSN: 1976-8605 (print).

* T. A. Naikoo, Bilal A. Rather and S. Pirzada, On Zagreb index of tournaments, accepted for publication in   Kragujevac Journal of Mathematics  ( Jan 2021).

Web of Science (ESCI), Scopus, ISSN: 2406-3045 (online), ISSN: 1450-9628 (print).

Communicated publications

* R. U. Shaban, Bilal A. Rather, S. Pirzada, On distance signless Laplacian spectral radius of power graphs of cyclic and dihedral groups, Submitted (Nov 2020).

* Bilal A. Rather, S. Pirzada, M Aouchiche, On eigenvalues and energy of geometric-arithmetic index of graphs, Submitted (Nov 2020).

* S. Pirzada, Bilal A. Rather, T. A. Naikoo, On general Zagreb index of tournaments, Submitted (Nov 2020).

* Bilal A. Rather, S. Pirzada, T. A. Naikoo, On Laplacian eigenvalues of zero divisor graphs of finite commutative ring, Submitted  (Oct 2020).

* Bilal A. Rather, Hilal A. Ganie and S. Pirzada, On the $ A_{\alpha} $-spectrum of joined union and its applications to power graphs of certain finite groups, Submitted  (Aug 2020).

* Hilal A. Ganie, Bilal A. Rather and S. Pirzada, Laplacian energy of trees of diameter four and beyond, Submitted (July 2020).

* Hilal A. Ganie, R. U. Shaban, Bilal A. Rather and S. Pirzada, On distance Laplacian energy, vertex connectivity and  independence number of graphs, Submitted (April 2020).

* S. Pirzada, Bilal A. Rather, Hilal A. Ganie and R. U. Shaban, On $ \alpha $-adjacency energy of graphs, Submitted (May 2020).

* S. Pirzada, Bilal A. Rather and T. A. Chishti , On normalized Laplacian spectrum of zero divisor graphs of commutative ring  $ \mathbb{Z}_{n} $, Submitted  (Febuary 2020).

* Bilal A. Rather, S. Pirzada and G. Zhou, On distance Laplacian spectra of power graphs of certain finite groups, Submitted  (March 2020).