On the $\mathcal{ABS}$ spectrum and energy of graphs
Swathi Shetty B. R. Rakshith Sayinath Udupa N. V

 

		On the 𝒜ℬ𝒮 spectrum and energy of  graphs  

		
 
	

	

		Swathi Shetty^1, B. R. Rakshith^*,2, Sayinath Udupa N. V. ^3

		Department of Mathematics, Manipal Institute of Technology
 Manipal Academy of Higher Education
 Manipal, India – 576104

		swathi.dscmpl2022@learner.manipal.edu^1

		ranmsc08@yahoo.co.in; rakshith.br@manipal.edu^*,2

		sayinath.udupa@manipal.edu	^3.
	

 § ABSTRACT 
Let η_1≥η_2≥⋯≥η_n be the eigenavalues of 𝒜ℬ𝒮 matrix. In this paper, we characterize connected graphs with 𝒜ℬ𝒮 eigenvalue η_n>-1. As a result, we determine all connected graphs with exactly two distinct 𝒜ℬ𝒮 eigenvalues. We show that a connected bipartite graph has three distinct 𝒜ℬ𝒮 eigenvalues if and only if it is a complete bipartite graph. Furthermore, we present some bounds for the 𝒜ℬ𝒮 spectral radius (resp. 𝒜ℬ𝒮 energy) and characterize extremal graphs. Also, we obtain a relation between 𝒜ℬ𝒞 energy and 𝒜ℬ𝒮 energy. Finally, the chemical importance of 𝒜ℬ𝒮 energy is investigated and it shown that the 𝒜ℬ𝒮 energy is useful in predicting certain properties of molecules. 
  AMS Classification:  05C50, 05C09, 05C35, 05C92.
  .

  
  Key Words: 𝒜ℬ𝒮 matrix, spectral radius, 𝒜ℬ𝒮 energy, QSPR analysis.


  

§ INTRODUCTION

  Throughout this article, we assume that G is a graph with vertex set V (G) and edge set E(G), where V(G)={v_1,v_2,…,v_n} and |E(G)|=m.   If two vertices v_i and v_j are adjacent, then
we write it as v_i∼ v_j otherwise, v_i≁v_j. We denote the degree of the vertex v_i by d(v_i)/d_i.
 Adjacency matrix of G is one of the well-studied graph matrix, denoted by A(G) and defined as A(G)=[a_ij]_n× n, where a_ij=1 if and only if v_i∼ v_j or 0, otherwise.   
 If λ_1≥λ_2≥⋯≥λ_n are the eigenvalues of A(G), then the sum  ∑_i=1^n |λ_i| is called the energy of graph G and is denoted by ℰ(G). The concept of graph energy, introduced by Gutman in 1978, slowly attracted mathematicians and chemists. In recent years, extensive research on graph energy has been carried out. For recent research on graph energy, see <cit.> and refer to the book "Graph Energy" by Li, Shi, and Gutman <cit.>. The study of graph energy is extended to various graph matrices, including (signless) Laplacian matrix, distance matrix, degree-based graph matrices and distance-based graph matrices. More than 50 graph energies have been defined so far. See <cit.> for more details.
  
A topological index is a numerical quantity derived from the graph's structure. In literature, plenty of topological indices are defined and used as molecular descriptors (see <cit.> ).  Most of the degree-based topological indices
 can be represented as TI(G)=∑_v_i∼ v_jℱ(d_i,d_j), where ℱ(d_i,d_j)= ℱ(d_j,d_i). As examples, we have first Zagreb index ℱ(d_i,d_j)=d_i+d_j, second Zagreb index ℱ(d_i,d_j)=d_id_j, Randić index (R(G)) ℱ(d_i,d_j)=1√(d_id_j), harmonic index (H(G)) ℱ(d_i,d_j)=2d_i+d_j, sum-connectivity index (χ(G)) ℱ(d_i,d_j)=1√(d_i+d_j), atom-bound connectivity index (𝒜ℬ𝒞(G)) ℱ(d_i,d_j)=√(d_i+d_j-2d_id_j), atom-bound sum-connectivity index (𝒜ℬ𝒮(G)) ℱ(d_i,d_j)=√(d_i+d_j-2d_i+d_j), etc.
 For a topological index TI(G), Das et al. <cit.> defined a general extended adjacency matrix as 𝒯=(t_ij)_n× n, where t_ij=ℱ(d_i,d_j) if v_i∼ v_j or 0, otherwise. The sum of  absolute values of all the eigenvalues of the matrix 𝒯 is called the energy of the general extended adjacency matrix 𝒯. In <cit.>, Das et al. obtained several lower and upper bounds for the energy of the matrix 𝒯, and deduced several known results about degree-based energies of graphs.

 The 𝒜ℬ𝒮 index  was introduced recently by Ali et al. in <cit.>. It combines both sum-connectivity index  and atom-bound sum connectivity index. Bounds on 𝒜ℬ𝒮 index for the classes of (molecular) trees and general graphs are obtained in <cit.> and also extremal graphs are classified.
 Chemical applicability of 𝒜ℬ𝒮-index is demonstrated in <cit.>. For more details about 𝒜ℬ𝒮 index  we refer to the survey article <cit.> by Ali et al. The 𝒜ℬ𝒮 matrix of G is defined to be the matrix 𝒜𝒮(G)=(w_ij)_n× n, where w_ij=√(d_i+d_j-2d_i+d_j) if v_i∼ v_j and 0, otherwise. We denote the eigenvalues of 𝒜𝒮(G) by η_1≥η_2≥⋯≥η_n. The sum ∑_i=1^n|η_i| is called the 𝒜ℬ𝒮 energy of G and is denoted by ℰ_ABS(G). The study of properties of 𝒜ℬ𝒮 matrix began recently. In <cit.>, it is proved that the 𝒜ℬ𝒮 Estrada index (∑_i=1^ne^η_i) of trees is maximum from the star graph and it is minimum for the path graph. Also, in <cit.>, the authors proved that 𝒜ℬ𝒮 spectral radius of a tree is maximum for star graph and it is minimum for the path graph. The chemical importance of the 𝒜ℬ𝒮 Estrada
 index and the 𝒜ℬ𝒮 spectral radius  are investigated separately in <cit.>, and it is shown that the 𝒜ℬ𝒮 Estrada index and  𝒜ℬ𝒮 spectral radius can be useful in  predicting  certain properties of molecules.

 Motivated by this, in Section 2 of the paper, we characterize connected graphs with 𝒜ℬ𝒮 eigenvalue η_n>-1. As a result, we determine all connected graphs with exactly two distinct 𝒜ℬ𝒮 eigenvalues. Further, we show that a connected bipartite graph has three distinct 𝒜ℬ𝒮 eigenvalues if and only if it is a complete bipartite graph. In Sections 3 and 4, we present some bounds for the 𝒜ℬ𝒮 spectral radius (resp. 𝒜ℬ𝒮 energy) and characterize extremal graphs. Also, we obtain a relation between 𝒜ℬ𝒞 energy and 𝒜ℬ𝒮 energy. In Section 5, the chemical importance of 𝒜ℬ𝒮 energy is investigated and it shown that the 𝒜ℬ𝒮 energy is useful to predict the boiling point and pi-electron energy of benzenoid hydrocarbons.

 As usual, the complete graph, path graph and  complete bipartite graph  on  n vertices are denoted by  K_n, P_n and K_n_1,n_2  (n_1+n_2=n), respectively.


§ PROPERTIES OF ABS EIGENVALUES

The following proposition is one of the basic properties of 𝒜ℬ𝒮 eigenvalues. We omit its proof as it is straightforward.
 
	Let G be a graph on n vertices. Let η_1≥η_2≥…≥η_n be its 𝒜ℬ𝒮- eigenvalues. Then
	 ∑_i=1^nη_i=0, ∑_i=1^n η_i^2=2(m-H(G)) and ∑_1≤ i<j≤ nη_iη_j=H(G)-m.
 Let M be a Hermitian matrix of order n. We denote the eigenvalues of M by θ_1(M)≥θ_2(M)≥⋯≥θ_n(M). The following lemma is the well-known Cauchy's interlacing theorem.
 <cit.>
 	Let M be a symmetric matrix of order n and let M_k be its leading principal k × k
 	submatrix. Then θ_n-k+i(M)≤θ_i(M_k)≤θ_i(M) for i=1,2,…,k.
 
 
 
	Let G be a graph on n vertices. Then the 𝒜ℬ𝒮 eigenvalues of G are all equal if and only if  G≅ p K_2∪ qK_1, where p+q=n.


	Suppose that the 𝒜ℬ𝒮 eigenvalues of G are all equal. Then by Proposition <ref>, ∑_i^n η_i=0, and so the   𝒜ℬ𝒮 eigenvalues of G are  zeros. Let H be a component of G. If |V(H)|≥ 3, then there exists a vertex x in H of degree at least two. Let y be a vertex of H adjacent to x. Then the principal minor of 𝒜𝒮(G) corresponding to the vertices x and y is non-zero. Thus, by Cauchy's interlacing theorem (see Lemma <ref>), the least eigenvalue of 𝒜𝒮(G) is non-zero, a contradiction. Hence, |V(H)|≤ 2. Therefore, G≅ pK_2∪ qK_1, where p+q=n. 
	Conversely, if G≅ pK_2∪ qK_1, then all the  entries of 𝒜𝒮(G) are zeros. Thus, η_1=η_2=…=η_n=0.

The diameter of a graph G is the maximum distance between any pair of vertices in G and it is denoted by diam(G). In the following theorem, we characterize connected graphs with η_n(G)> -1.

Let G be connected graph on n vertices. Then η_n(G)> -1 if and only if G≅ K_n or P_3.	





Assume that diam(G)≥ 2 and  G P_3. Let x-y-z be an induced path in G.  
Then either d(y)≥ 3, d(x)≥ 2 or d(z)≥ 2. Let 𝒜𝒮[p,q,r] denote the principal submatrix of 𝒜𝒮(G) corresponding to the vertices p,q and r, where p-q-r is an induced path in G. Let θ_1[p-q-r]≥θ_2[p-q-r]≥θ_3[p-q-r] be the  eigenvalues of 𝒜𝒮[p,q,r]. Then 

    θ_1[p-q-r]   =   √(2)  √(1-1d(q)+d(r)-1d(q)+d(p));
    θ_2[p-q-r]   =   0;
    θ_3[p-q-r]   =   -√(2)  √(1-1d(q)+d(r)-1d(q)+d(p)).

Also, by Cauchy's interlacing theorem (see Lemma <ref>), η_n(G)≤θ_3[p-q-r].

Case 1: Let d(y)≥ 3, d(x)≥1 and d(z)≥1 (see Fig. <ref>(b)). Then
 2(1-1d(y)+d(z)-1d(y)+d(x))≥ 1. Thus, 
η_n(G)≤θ_3[x-y-z]≤ -1.

Case 2: Let d(y)=2, d(x)≥2 and d(z)≥1. If d(z)≥ 2, then
 2(1-1d(y)+d(z)-1d(y)+d(x))≥ 1. So, η_n(G)≤θ_3[x-y-z]≤ -1. Otherwise, d(z)=1. Let s be a vertex adjacent with the vertex x in G (see Fig. 1(c)). If G≅ P_4, then η_n(G)=-1.0306<-1. Suppose G P_4, then either d(x)≥3 or d(s)≥ 2.

Subcase 2.1: Let d(x)≥2 and d(s)≥ 2 (see Fig. <ref>(d)). Then
 2(1-1d(s)+d(x)-1d(y)+d(x))≥ 1. Thus, η_n(G)≤θ_3[s-x-y]≤ -1.

Subcase 2.2: Let d(x)≥ 3 and d(s)=1. Then there exists a vertex r adjacent with the vertex x in G (see Fig. 1(e)).Therefore, 2(1-1d(s)+d(x)-1d(r)+d(x))≥ 1. Thus, η_n(G)≤θ_3[s-x-r]≤ -1.

Thus for a connected graph G P_3 with diam(G)≥ 2, η_n(G)≤ -1. Hence G≅ K_n or P_3.  Conversely, η_n(K_n)=-√(n-2n-1)>-1 and η_3(P_3)=-0.81649>-1. This completes the proof of the theorem. 


Let G be a connected graph of order n≥2. Then η_n=-√(n-2n-1) if and only if G≅ K_n.


Suppose η_n=-√(n-2n-1). Then by Theorem <ref>, G≅ K_n or P_3. Since n_3(P_3)=-√(23) (≠ -√(12)), G≅ K_n. The converse part is direct. 

Let B_1 and B_2 be two real matrices of same order, we write B_1≼ B_2 if every entry in B_1 does not exceed the counterpart in B_2. The following lemma is useful to prove our next result. <cit.>
	Let B_1, B_2 be non-negative matrices of order n. If B_1≼ B_2, then ρ(B_1)≤ρ(B_2). Further, if B_1 is irreducible and B_1≠ B_2, then ρ(B_1)< ρ(B_2).



Let G be a connected graph of order n>2. Then 𝒜𝒮(G) has exactly two distinct eigenvalues if and only if G≅ K_n.


Suppose G has exactly two distinct eigenvalues. Since G is a connected graph of order n> 2, the matrix 𝒜𝒮(G) is irreducible, and thus by Perron-Frobenius theory its largest eigenvalue, i.e., η_1(G) is a simple eigenvalue of G. So, η_1(G) and η_n(G) are the two distinct eigenvalues of G, and η_2(G)=η_3(G)=⋯=η_n(G).  
Let B=√(n-2n-1) A(K_n). Then the eigenvalues of B are (n-1)√(n-2n-1),-√(n-2n-1),…,-√(n-2n-1)_n-1. Since 𝒜𝒮(G)≼ B, η_1(G)≤ (n-1)√(n-2n-1) by Lemma <ref>. Therefore, -(n-1)η_n(G)≤(n-1)√(n-2n-1). That is, η_n(G)≥ -√(n-2n-1)>-1. Hence by Theorem <ref>, G≅ K_n or P_3. So, G≅ K_n because P_3 has three distinct eigenvalues. The converse part is straightforward.    

The following lemma is important to prove our next result.
<cit.>
	Let C∈ M_n,m, q=min{n,m},  σ_1≥σ_2 ≥…≥σ_q be the ordered singular values of C, and define the Hermitian matrix ℋ=[   0   C; C^*   0 ]. The ordered eigenvalues of ℋ are -σ_ 1≤…≤ -σ_q≤ 0=…= 0≤σ_q≤…≤σ_1.


Let G be a graph of order n. Let M=(m_ij)_n× n be a non-negative symmetric matrix of order n, where m_ij is positive if and only if v_i∼ v_j. Then the graph G is bipartite if and only if the eigenvalues of the matrix M are symmetric about origin.

Suppose G is a  bipartite graph. Then the matrix M can be written as M=[   0   C; C^T   0 ], where C is a rectangular matrix with non-negative entries. Therefore, by Lemma <ref>, the eigenvalues of M are symmetric about the origin.

Conversely, suppose the eigenvalues of M are symmetric about the origin. Then trace(M^k)=0 for all odd integer k>0.  By the definition of the matrix M, one can easily check that trace(M^k(G))>0 if and only if trace(A^k(G))>0. Now, assume that G contains an odd cycle of length k. Then trace(A^k(G))>0 (see <cit.>). So, trace(M^k)>0, a contradiction. Thus, G must be a bipartite graph.  

The following corollary is immediate from the above theorem

A graph G is bipartite if and only if the eigenvalues of 𝒜𝒮(G) are symmetric about origin.


A connected bipartite graph G of order n>2 has three distinct 𝒜ℬ𝒮 eigenvalues if and only if G is a complete bipartite graph.


From Perron Frobenius theorem and by Corollary <ref>, η_1(G)>0 and η_n(G) are simple 𝒜ℬ𝒮 eigenvalues of G. Suppose G has a non-zero 𝒜ℬ𝒮 eigenvalue other than η_1(G) and η_n(G). Then by Corollary <ref>, G must have at least four distinct 𝒜ℬ𝒮 eigenvalues. Thus, 0 is an 𝒜ℬ𝒮 eigenvalue of G with multiplicity n-2.  Hence, rank(𝒜𝒮(G))=2. Let U and W be the vertex partition sets of G. Let u∈ U and w∈ W. Since G is a connected bipartite graph, the rows corresponding to the vertices u and w are linearly independent. Further, since rank(𝒜(𝒮)(G))=2 and G is bipartite, the rows corresponding to the vertices belonging to U(respectively, W) are in the linear span of the row vector corresponding to the vertex u (respectively, w). Thus, the vertices in U (respectively, W) share the same vertex neighborhood set.
Assume that w∈ W and w∉ N(u). Then w is not adjacent with any vertices of U. Therefore, w is an isolated vertex of G, a contradiction because G is a connected graph of order at least 3. Thus, N(u)=W and N(w)=U. Therefore, G is a complete bipartite graph.

Conversely, if G is the complete bipartite graph K_n_1,n_2 of order n=n_1+n_2 (≥ 3), then 𝒜𝒮(G)=√(1-2n_1+n_2)A(G). Therefore the 𝒜ℬ𝒮 eigenvalues of G are
 √(n_1n_2(1-2n_1+n_2)),0,0,…,0_n-2, -√(n_1n_2(1-2n_1+n_2)). Thus, G has exactly three distinct 𝒜ℬ𝒮 eigenvalues.



§ BOUNDS FOR THE ABS SPECTRAL RADIUS

 In this section, we give some bounds for the largest 𝒜ℬ𝒮 eigenvalue η_1(G).
 
Let G be graph of order n with maximum degree Δ and minimum degree δ. Then the row sums of 𝒜𝒮(G) are equal if and only if G is a regular graph.


Suppose the row sums of 𝒜𝒮(G) are equal . Let u,v∈ V(G) such that d(u)=δ and d(v)=Δ. Then
    ∑_v_i:u∼ v_i√(1-2δ+d_i)=∑_v_j:v∼ v_j√(1-2Δ+d_j).
If δ≠Δ, then ∑_v_j:v∼ v_j√(1-2Δ+d_j)≥Δ√(1-2Δ+δ) >∑_v_i:u∼ v_i√(1-2δ+d_i), a contradiction to the equation (<ref>). Thus, δ=Δ. i.e., G is a regular graph. The converse part is straightforward.
 
 The following theorem gives a lower bound for η_1(G) in terms of order and the atom-bond sum connectivity index of graph G.

Let G be a graph of order n, minimum degree δ and maximum degree  Δ. Then 
   η_1(G) ≥2𝒜ℬ𝒮(G)n. Further, equality holds if and only if G is a regular graph.


    Let x=(x_1,x_2,…,x_n)^T be a vector in ℝ^n. Then

        x^T𝒜𝒮(G)x=2∑_v_i∼ v_j√(d_i+d_j-2d_i+d_j)x_ix_j.
    
  
Set x=(1,1,…,1)^T.  Then by Rayleigh's inequality, η_1(G)≥x^T𝒜𝒮(G) xx^Tx=2𝒜ℬ𝒮(G)n, and the equality holds if and only if x=(1,1,…,1)^T is an eigenvector of 𝒜𝒮(G) corresponding to the eigenvalue η_1(G). Suppose η_1(G)=2𝒜ℬ𝒮(G)n. Then the row sums of  𝒜𝒮(G) are equal. Therefore, by Lemma <ref>, G is a regular graph.

Next, we provide a lower and upper bound for η_1(G) in terms of order, size and the harmonic index of graph G.

	Let G be a graph of order n  and size m with no isolated vertices. Then 
    √(2(m-H(G))/n)≤η_1(G)≤√(2(n-1)/n(m-H(G)))

	with equality holds if and only if n is even and G≅n2K_2.   


	By Proposition <ref>, 
    nη_1^2≥∑_i=1^n η_i^2=2(m-H(G)).

 Since ∑_i=1^n η_i=0, η_1^2=(∑_i=2^n η_i)^2. Therefore by Cauchy-Schwarz inequality,
   η_1^2≤ (n-1)∑_i=2^n η_i^2 and the equality holds if and only if η_2=η_3=⋯=η_n.
 So,
    nη_1^2    ≤ 2(n-1)(m-H(G)).

 Thus from equations (<ref>) and (<ref>) we get the desired inequality.  
	Suppose n is even and G≅n2K_2. Then 𝒜𝒮(G) is the null matrix, and so η_1=η_2=…=η_n=0 and H(G)=m. Thus the equalities in (<ref>) holds.
	Conversely, suppose the right equality holds. Then  from equation (<ref>), η_1^2=η_2^2=…=η_n^2. This implies that G has at most two distinct 𝒜ℬ𝒮 eigenvalue. Therefore, by Theorems <ref> and <ref>, either G≅ K_n (n>2) or n is even and G≅n2K_2. If G≅ K_n, then η_1^2≠η_2^2, a contradiction. Thus, G≅n2K_2 for an even integer n. Similarly, the left equality holds if and only if G≅n2K_2 for an even integer n.


    Let G be a graph of order n with minimum degree δ and maximum degree Δ. Then √(δ(δ-1))≤η_1(G)≤√(Δ(Δ-1)). Further, equality holds if and only if G is a regular graph.


From <cit.>, min_1≤ i≤ n{R_i}≤η_1(G)≤max_1≤ i≤ n{R_i}, where R_i is the row sum of the ith row of 𝒜𝒮(G). Moreover, the equality on both sides holds if and only if all the row sums of 𝒜𝒮(G) are equal. Now, 

    max_1≤ i≤ n{R_i}=max_1≤ i≤ n ∑_v_i:v_i∼ v_j√(1-2/d_i+d_j)   ≤√(Δ(Δ-1)),
 
where the equality holds if and only if one of the components of G is a Δ-regular graph,
and

    min_1≤ i≤ n{R_i}=	min_1≤ i≤ n ∑_v_i:v_i∼ v_j√(1-2/d_i+d_j)   ≥√(δ(δ-1)),
 
where the equality holds if and only if one of the components of G is a δ-regular graph. Now, by Lemma <ref>, the row sums of 𝒜𝒮(G) is a constant if and only if G is regular. Thus, 
√(δ(δ-1))≤η_1(G)≤√(Δ(Δ-1)) and the equality on both sides holds if and only if G is a regular graph.

The sum connectivity matrix of a graph G is a general extended adjacency matrix with ℱ(d_i,d_j)=1√(d_i+d_j). It is denoted by 𝒮(G). In the following theorem, we give a relation between the spectral radius of 𝒮(G) (ρ(𝒮(G))) and η_1(G).

	If G is a connected graph of order n≥2 , then 

		ρ(𝒮(G)) min_
v_i∼ v_j√(d_i+d_j-2)≤η_1(G)≤ρ(𝒮(G))max_
v_i∼ v_j√(d_i+d_j-2). 
	
 Further, equality on both sides holds if and only if G is a regular graph or semiregular graph.


	The matrices 𝒜𝒮(G) and 𝒮(G) are non-negative and irreducible. Moreover, 
    𝒮(G)min_
    v_i∼ v_j√(d_i+d_j-2)≼𝒜𝒮(G)≼𝒮(G)max_
    v_i∼ v_j√(d_i+d_j-2).

	Thus, by Lemma <ref>,
    ρ(𝒮(G))min_
    v_i∼ v_j√(d_i+d_j-2)≤η_1(G)≤ρ(𝒮(G))max_
    v_i∼ v_j√(d_i+d_j-2).

Now we consider the equality case. Suppose η_1(G)=ρ(𝒮(G))max_v_i∼ v_j√(d_i+d_j-2). Then by Lemma <ref>, 𝒜𝒮(G)=𝒮(G)max_v_i∼ v_j√(d_i+d_j-2). This implies that, for every edge v_iv_j in G, √(d_i+d_j-2)=max_v_i∼ v_j√(d_i+d_j-2). Therefore, d_i+d_j is constant for any edge v_iv_j of G.  Let u (resp. v) be a vertex in G with maximum degree Δ  (resp. minimum degree δ). Let u∼ u_1 and v∼ v_1. Then d(u)+d(u_1)≥Δ+δ≥ d(v)+d(v_1). Hence, d_i+d_j=δ+Δ, for all v_iv_j∈ E(G). Now, suppose there exists a vertex w in G such that d(w)∈(δ, Δ). Then G has a component whose vertex degrees are either d(w) or Δ+δ-d(w). Therefore, G is disconnected,  a contradiction.  Thus, G is Δ-regular or (δ, Δ)-semiregular graph. Similarly, if η_1(G)=ρ(𝒮(G))min_v_i∼ v_j√(d_i+d_j-2), then G is Δ-regular or (δ, Δ)-semiregular graph. The converse part is straightforward.



    If G is a connected graph with maximum degree Δ and minimum degree δ, then 2χ(G)n√(2δ-2)≤η_1(G)≤√(2(Δ-1)(n-1)n R(G)), where the equality on the left side holds only if G is regular and the equality on the right side holds only if G is a complete graph.


From <cit.>, we have

    2χ(G)n≤ρ(𝒮(G))≤√(n-1nR(G)),

where the left side equality holds only if  𝒮(G) has equal row sums, and the right equality holds only if G is a complete graph.   Therefore by Theorem <ref>, we get the desired result.	

Now, we obtain an upper bound for the spectral radius of unicyclic graphs using the following lemma.
<cit.>
	Let P≥ 0 be an irreducible matrix with an eigenvalue θ.If PY≤ tY for t∈ℝ, Y∈ℝ^n and Y≥ 0, then t≥θ.


	Let G be a unicyclic graph of order n and girth at least 5. Then η_1(G)≤√((n-3)(n+1)n-1).


	Since G is unicyclic of order n,  ∑_i=1^n d_i=2n.
 Therefore, 
    ∑_v_j:v_i∼ v_j d_j   =2n-d_i-∑_v_j:v_j≁v_i  d_j≤ 2n-d_i-∑_v_j:v_j≁v_i 1 =n+1.
 
	That is, ∑_v_j:v_i∼ v_j d_j≤ n+1. Also, d_i+d_j=2n-∑_k≠ i, jk=1^nd_k. Since G is of girth at least 5, there exist at least 3 vertices v_p,v_q and v_r on the cycle which are distinct from the vertices v_i and v_j. Therefore, d_i+d_j≤ 2n-6-∑_k∉{i,j,p,q,r}d_k≤ n-1. That is, d_i+d_j≤ n-1. 
	Let Δ_a=max{d_i+d_j: v_iv_j∈ E(G)},  Y={√(d_1),√(d_2),…,√(d_n)} and (𝒜𝒮Y)_i denote the i^th row sum of the matrix (𝒜𝒮)Y. 
	Then Δ_a≤ n-1, and so 
	

		(𝒜𝒮Y)_i=∑_v_j:v_i∼ v_j√(1-2d_i+d_j)√(d_j)≤√(1-2n-1)∑_v_j:v_i∼ v_j√(d_j).

	Now by Cauchy-Schwarz inequality, 

		∑_v_j:v_i∼ v_j√(d_j)≤√(∑_v_j:v_i∼ v_jd_j)√(d_i)≤√(n+1)√(d_i).

	Therefore, (𝒜𝒮Y)_i≤√((n-3)(n+1)n-1)√(d_i), and thus
	𝒜𝒮Y≤√((n-3)(n+1)n-1)  Y.
	Hence, by Lemma <ref>, η_1(G)≤√((n-3)(n+1)n-1).



§  PROPERTIES OF ATOM-BOND SUM-CONNECTIVITY ENERGY

In this section, we present some bounds on ℰ_𝒜ℬ𝒮(G).

	Let G be a graph with n ≥ 2 vertices and m edges. Then 
	
  * ℰ_𝒜ℬ𝒮(G)≥2√(m-H(G)). Equality holds if and only if  G≅ pK_n_1,n_2∪ qK_2∪ rK_1, where n_1+n_2>2,  p=0 or 1 and p(n_1+n_2)+2q+r=n.
	 
  * ℰ_𝒜ℬ𝒮(G)≤√(2n(m-H(G))). Equality holds if and only if G≅ pK_2∪ qk_1, where 2p+q=n.
	 

 (i) From <cit.>, we have ℰ_𝒜ℬ𝒮(G)≥√(2trace(𝒜𝒮^2(G))) and the equality holds if and only if η_1=-η_n and η_2=η_3=⋯=η_n-1=0. Since trace(𝒜𝒮^2(G))=2m-H(G), we get ℰ_𝒜ℬ𝒮(G)≥2√(m-H(G)).  Suppose η_1=-η_n and η_2=η_3=⋯=η_n-1=0. Then G is bipartite  (see, Corollary <ref>). Also, if H is a component of G, then 𝒜𝒮(H) has either two or three  distinct eigenvalues, or all its eigenvalues are equal to 0. Thus, from Theorems <ref>, we get H≅ K_n_1,n_2 with n_1+n_2>2, K_2 or K_1. Furthermore, if K_n_1,n_2 (n_1+n_2>2) is a component of G, then all other components of G are either K_2 or K_1. Otherwise η_2>0, a contradiction.

(ii) From<cit.>, we have ℰ_𝒜ℬ𝒮(G)≤√(n  trace(𝒜𝒮^2(G))) and the equality holds if and only of |η_1|=|η_2|=⋯=|η_n|. Therefore, ℰ_𝒜ℬ𝒮(G)≤√(2n (m-H(G))). Suppose |η_1|=|η_2|=⋯=|η_n|. Then the eigenvalues of 𝒜𝒮(G) are all equal or it has exactly two distinct eigenvalues. So, by Theorems <ref> and <ref>, each component of G is K_n_1 for some positive integer n_1. Furthermore, n_1=1 or 2. Otherwise η_1>η_2. This completes the proof. 

To prove our next upper bound on ℰ_𝒜ℬ𝒮(G), we need the following lemma.
<cit.>
A regular connected graph G is strongly regular if and only if it has three distinct eigenvalues.


	Let G be a graph of order n with m edges. 
    If m=H(G) or 2(m-H(G))≥ n, then ℰ_𝒜ℬ𝒮(G)≤2𝒜ℬ𝒮(G)n+√((n-1)(2m-2H(G)-4 (𝒜ℬ𝒮(G))^2n^2)). Further, equality holds if and only if G≅ pK_2∪ qK_1, where 2p+q=n, G≅ K_n or G is a non-complete strongly regular graph.


     Using Cauchy-Schwarz inequality, 
    ℰ_𝒜ℬ𝒮(G)=∑_i=1^n |η_i|   =η_1+∑_i=2^n |η_i|
            ≤η_1+√((n-1)(2m-2H(G)-η_1^2)),
 where the equality holds if and only if |η_2|=|η_3|=⋯=|η_n|. Let g(x)=x+√((n-1)(2m-2H(G)-x^2)). Then by first derivative test, the function g is decreasing for √(2(m-H(G))n)≤ x≤√(2(m-H(G))).  
Now, 2𝒜ℬ𝒮(G)n=2n∑_v_iv_j∈ E(G)√(d_i+d_j-2d_i+d_j)≥2n∑_v_iv_j∈ E(G)d_i+d_j-2d_i+d_j=2(m-H(G))n≥√(2(m-H(G))n) (because 2(m-H(G))≥ n). That is, 2𝒜ℬ𝒮(G)n≥√(2(m-H(G))n).  Upon combining the above inequality with Theorem <ref>, we get

    √(2(m-H(G))n)≤2 𝒜ℬ𝒮(G)n≤η_1≤√(2(m-H(G))).

 Therefore, 
    ℰ_𝒜ℬ𝒮(G)≤ g(η_1)≤ g(2𝒜ℬ𝒮(G)n)= 2𝒜ℬ𝒮(G)n+√((n-1)(2m-2H(G)-4 (𝒜ℬ𝒮(G))^2n^2))
    .

  Suppose the equality in equation (<ref>) holds. Then η_1=2𝒜ℬ𝒮(G)n and |η_2|=|η_3|=⋯=|
    η_n|. Thus, 𝒜𝒮(G) has at most three distinct eigenvalues. If  𝒜𝒮(G) has at most two distinct eigenvalues, then by Theorems <ref> and <ref>, G≅ pK_2∪ qK_1, where p+q=n, or G≅ K_n. Otherwise, 𝒜𝒮(G) has exactly three distinct eigenvalues. Now, by Theorem <ref>, G is  k-regular graph for some constant k, and so 𝒜𝒮(G)=√(k-1k)A(G). If 𝒜𝒮(G) has exactly three distinct eigenvalues, then G has exactly three distinct  eigenvalues. Therefore by Lemma <ref>, G must be a non-complete strongly regular graph. Conversely, if G≅ pK_2∪ qK_1, where 2p+q=n, or G≅ K_n, then one can easily see that the equality in (<ref>) holds. Suppose G is a non-complete strongly k-regular graph, then η_1=√(k(k-1)), |η_j|=√((n-k)(k-1)n-1) for j=2,3,…,n. Now, one can easily check that equality in (<ref>) holds. This completes the proof of the theorem. 
     
The following lemmas are useful to prove our next result.

<cit.>
	If M is a Hermitian n× n matrix, then |θ_1(M)-θ_n(M)|≥ 2max_j(∑_k:k≠ j|a_jk|^2)^1/2.



	Let G be a connected graph of order n≥ 2 with maximum degree Δ and minimum degree δ. Then
	

		2√(Δ(Δ+δ-2)Δ+δ)≤η_1+|η_n|≤  2√(m-H(G)).



	By Lemma <ref>, 
    η_1+|η_n|≥ 2 max_i(∑_v_j:v_i∼ v_jd_i+d_j-2/d_i+d_j)^1/2≥ 2√(Δ(Δ+δ-2)/Δ+δ).

	Proving the left inequality.
	Now, by Cauchy-Schwarz inequality and from Proposition <ref>, 
    η_1+|η_n|≤√(2(η_1^2+η_n^2))≤√(4(m-H(G)))=2√(m-H(G)).



Let G be  (n, m)-graph with maximum degree Δ.  	
If 2(m-H(G))n≤Δ(Δ-1)Δ+1, ℰ_𝒜ℬ𝒮(G)≤ 2√(Δ(Δ-1)Δ+1)+√(2(n-2)(m-H(G)-Δ(Δ-1)Δ+1)). Equality holds if and only if G≅ pK_1,Δ∪ qK_2∪ rK_1, where p=0 or 1 and p(Δ+1)+2q+r=n.


Let η_1≥η_2≥⋯≥η_n be the 𝒜ℬ𝒮 eigenvalues of G. Using Cauchy-Schwarz inequality, 
    
    ℰ_𝒜ℬ𝒮(G)=η_1+|η_n|+∑_i=2^n-1|η_i|≤η_1+|η_n|+√(∑_i=2^n-1(n-2)|η_i|^2),

   where the equality holds if and only if |η_2|=|η_3|=⋯=|η_n-1|.
   Therefore, by Proposition <ref>,
   
    ℰ_𝒜ℬ𝒮(G)    ≤η_1+|η_n|+√((n-2)(2m-2H(G)-η_1^2-η_n^2)).

Further, by A.M.-G.M. inequality, 2√(η_1η_n)≤η_1+|η_n| and the equality holds if and only if η_1=|η_n|, Thus

    ℰ_𝒜ℬ𝒮(G)    ≤η_1+|η_n|+√((n-2)(2m-2H(G)-(η_1+|η_n|)^2/2)).

 Let f(x)=2x+√((n-2)(2m-2H(G)-2x^2)). Then f is decreasing for √(2(m-H(G))n)≤ x≤√(m-H(G)). By Lemma <ref>, 
    √(2(m-H(G))/n)≤√(Δ(Δ-1)/Δ+1)≤η_1+|η_n|/2≤√(m-H(G)).
 So,
    E_𝒜ℬ𝒮(G)   ≤ f(η_1+|η_n|/2)≤ f(√(Δ(Δ-1)/Δ+1))
       =2√(Δ(Δ-1)/Δ+1)+√((n-2)(2m-2H(G)-2Δ(Δ-1)/Δ+1)).
 
Suppose the equality in equation (<ref>) holds. Then η_1=|η_n|=√(Δ(Δ-1)Δ+1) and |η_2|=|η_3|=⋯=|η_n-1|. Thus, by Perron-Frobenius theorem, G is a bipartite graph. 
If Δ=1, then, G≅ qK_2∪ rK_1, where 2q+r=n. Otherwise, Δ>1, and so η_1=|η_n|>0 . Let H be component of G having a vertex of degree Δ. Since G is bipartite, K_1,Δ is an induced subgraph of H. So, by Cauchy's interlacing theorem,
η_1(H)≥η_1(K_1,Δ)=√(Δ(Δ-1)Δ+1).  Moreover the equality holds if and only if H≅ K_1,Δ.  Now, η_1=√(Δ(Δ-1)Δ+1)≥η_1(H) because H is a component of G. Therefore, η_1(H)=η_1=Δ(Δ-1)Δ+1 and H≅ K_1,Δ. Further,   0 is an 𝒜ℬ𝒮 eigenvalue of H, and thus  𝒜ℬ𝒮 eigenvalues of G are η_1=|η_n|=√(Δ(Δ-1)Δ+1) and |η_2|=|η_3|=⋯=|η_n-1|=0. Therefore, if H_1 H is a component of G, then all its 𝒜ℬ𝒮 eigenvalues are equal to 0. Therefore, by Theorem <ref>, H_1≅ K_2 or K_1. Thus G≅ pK_1,Δ∪ qK_2∪ rK_1, where p=0 or 1 and p(Δ+1)+2q+r=n. Conversely, if  G≅ pK_1,Δ∪ qK_2∪ rK_1, where p=0 or 1 and p(Δ+1)+2q+r=n, then one can easily verify that the equality holds.


    Let φ_1≥φ_2≥⋯≥φ_n be the 𝒜ℬ𝒞-eigenvalues and η_1≥η_2≥⋯≥η_n be the 𝒜ℬ𝒮 eigenvalues of a graph G without pendent vertices. Then ℰ_𝒜ℬ𝒮(G)≥√(2n)ℰ_𝒜ℬ𝒞(G).


    We have, 
   
    (ℰ_𝒜ℬ𝒮(G))^2   =(∑_i=1^n |η_i|)^2=∑_i=1^n η_i^2+2∑_i<j|η_i||η_j| 
       ≥∑_i=1^n η_i^2+2 |∑_i<jη_iη_j|  (by triangle inequality)
       = 2 ∑_i=1^n η_i^2    (because∑_i=1^n η_i^2=-2 ∑_i<jη_iη_j)
       =4∑_v_iv_j∈ E(G)d_i+d_j-2d_i+d_j
       ≥ 4∑_v_iv_j∈ E(G)d_i+d_j-2d_id_j
       =2∑_i=1φ_i^2≥2n(∑_i=1|φ_i|)^2 (by Cauchy-Schwarz inequality)
       =2n(ℰ_𝒜ℬ𝒞(G))^2.
 
Thus, ℰ_𝒜ℬ𝒮(G)≥√(2n)ℰ_𝒜ℬ𝒞(G).

 

§ QSPR ANALYSIS OF BENZENOID HYDROCARBON

In this section, we show that the physicochemical properties, namely, the boiling point (BP) and pi-electron energy (ℰ_π)of benzenoid hydrocarbons can be modeled using 𝒜ℬ𝒮-energy.
 The experimental values listed in this section are taken from <cit.>. The hydrogen-suppressed molecular graphs are depicted in Figure <ref>. 

 The calculated values of 𝒜ℬ𝒮 energy for benzenoid hydrocarbons are shown in Table <ref> . 

Consider the following model:

    Y= A (± S_e)ℰ_𝒜ℬ𝒮+B(± S_e),

 where Y, A, S_e and B denote the property, slope, standard error of coefficients and intercept, respectively. We denote  the correlation coefficient,  standard error of the model, the F-test value  and the significance by r, SE, F and SF, respectively.
 For benzenoid hydrocarbons, it is found that the 𝒜ℬ𝒮 energy has a strong correlation with the boiling point and pi-electron energy. In fact, we get the following regression equations for benzenoid hydrocarbons using model (<ref>).

    BP=22.684(± 0.4013)ℰ_𝒜ℬ𝒮+2.25(± 8.7952),

                 r^2=0.9941,  SE=7.8838,  F=3194.002,  SF=1.23× 10^-22.


    ℰ_π=-1.2668(± 0.01235)ℰ_𝒜ℬ𝒮+1.0586(± 0.2706),

                r^2=0.9982,  SE=0.2425, F=10522.29,  SF=1.54× 10^-27.

The data variance for BP and pi-electron energy is around 99%. The standard errors are very low, particularly in model (<ref>), where they are significantly small. This low standard error enhances the model's consistency and increases the F-value, especially for pi-electron energy. The SF values are significantly below 0.05.  The predicted properties from model (<ref>) are compared with the experimental properties using bar diagrams, where series 1 is related to experimental value and series 2 is related to predicted value. These figures show that the experimental and predicted data align well. Additionally, the residuals are randomly scattered around the zero line, indicating that the model is consistent.

In <cit.> the QSPR analysis of benzenoid hydrocarbons is done using the second-degree based entropy, ve-degree irregularity index, Albertson index, first and second status connectivity indices, first and second eccentric connectivity indices, Wiener index, Sombor index, reduced Sombor index and 𝒜ℬ𝒮 index. It is observed that the |r| value obtained for BP using ℰ_𝒜ℬ𝒮 is better than that of |r| value obtained from these indices. Further, ℰ_𝒜ℬ𝒮 have high pi-electron energy predictive ability compared to second-degree based entropy and 𝒜ℬ𝒮 index. With the smaller standard error and higher F-value of the proposed models, we can conclude that the performance of the models is better than that of the models discussed in <cit.> using 𝒜ℬ𝒮 index.


§ CONCLUSIONS

In this work, we have determined all connected graphs with η_n>-1. As a result, graphs with two distinct 𝒜ℬ𝒮 eigenvalues are classified. Also, bipartite graphs with three distinct 𝒜ℬ𝒮 eigenvalues are determined. Further, some bounds on the spectral radius and energy of the matrix 𝒜𝒮(G) are obtained. The problem of characterizing non-bipartite graphs with three distinct 𝒜ℬ𝒮 eigenvalues remains open. Also, the chemical importance of 𝒜ℬ𝒮 energy is demonstrated. As a future work, the problem of obtaining sharp bounds for the spectral radius and energy of the matrix 𝒜𝒮(G) in terms of graph parameters would be interesting.     
99
	akbariarelations S. Akbari, M. Habibi, S. Rabizadeh, Relations between Energy and Sombor Index,
	MATCH Commun. Math. Comput. Chem. 92(2024) 425–435.
	alawiah2018new N. Alawiah, N. J. Rad, A. Jahanbani, H. Kamarulhaili. New upper bounds on the energy of a graph, MATCH Commun. Math. Comput. Chem. 79(2018) 287–301.
	ali2020symmetric A. Ali, S. Elumalai, T. Mansour, On the symmetric division deg index of molecular graphs, MATCH Commun. Math. Comput. Chem.  83 (2020) 205–220.
	ali2022atom A. Ali, B. Furtula, I.Redžepović, I. Gutman, Atom-bond sum-connectivity
	index, Journal of Mathematical Chemistry. 60 (2022) 2081–2093.
	ali2024extremal A. Ali, I. Gutman, B. Furtula, I. Redžepović, T. Došlić, Z. Raza, Extremal
	results and bounds for atom-bond sum-connectivity index, MATCH Commun. Math. Comput. Chem. 92(2024) 271–314.
	ali2023atom A. Ali, I. Gutman, and I.Redžepović, Atom-bond sum-connectivity index of unicyclic
	graphs and some applications, Electron. J. Math. 5(2023) 1–7.
	brouwer2011spectra A. E. Brouwer, W. H. Haemers, Spectra of graphs, Springer, New York, 2011.
	cvetkovic1980spectra D. M. Cvetković, M. Doob, H. Sachs, Spectra of graphs: theory and applications, Academic Press, New York, 1980.
	tavakoli2024energy K. C. Das, G. Ali, M. Tavakoli, On the Energy and Spread of the Adjacency, Laplacian and Signless Laplacian Matrices of Graphs, MATCH Commun. Math. Comput. Chem. 92(2024) 545–566.
	das2018degree K. C. Das, I. Gutman, I. Milovanović, E. Milovanović, B. Furtula, Degree-based
	energies of graphs, Linear algebra and its applications. 554(2018) 185–204.
	das2022ve K. C. Das, S. Mondal, On ve-degree irregularity index of graphs and its applications as molecular descriptor, Symmetry. 14(2022) 2406.
	das2023neighborhood K. C. Das, S. Mondal, On neighborhood inverse sum indeg index of molecular
	graphs with chemical significance, Information Sciences. 623(2023) 112–131.
	espinalgraph C. Espinal, J. Rada, Graph energy change due to vertex deletion, MATCH Commun. Math. Comput. Chem. 92(2024) 89–103.
	estrada2017meaningE. Estrada, M. Benzi, What is the meaning of the graph energy after all?, Discrete
		Applied Mathematics. 230(2017) 71–77.
	gutman2013degreeI. Gutman, Degree-based topological indices, Croatica chemica acta. 86(2013) 351–361.
	gutman2021geometric I. Gutman, Geometric approach to degree-based topological indices: Sombor indices,
	MATCH Commun. Math. Comput. Chem. 86(2021) 11–16.
	gutman2020research I. Gutman, H. Ramane, Research on graph energies in 2019,MATCH Commun. Math. Comput. Chem.  84(2020) 277–292.
	horn2012matrixR. A. Horn, C. R. Johnson, Matrix analysis, Cambridge university press, 2012.
	li2012graph X. Li, Y. Shi, I. Gutman, Graph energy, Springer Science & Business Media, 2012.
	lin2024abs Z. Lin, T. Zhou,Y. Liu, On ABS Estrada index of trees,  Journal of Applied
		Mathematics and Computing.(2024) 1–13.
	lin2024abstrees Z. Lin, T. Zhou,Y. Liu, On the atom-bond sum-connectivity spectral radius of trees, Discrete Mathematics Letters.(2024) 122–127.
	liu2021moreH. Liu, H. Chen, Q. Xiao, X. Fang, Z. Tang, More on Sombor indices of chemical graphs and their applications to the boiling point of benzenoid hydrocarbons,
	International Journal of Quantum Chemistry. 121(2021) 26689.
	merikoski2003characterizationsJ. K. Merikoski, R. Kumar, Characterizations and lower bounds for the spread
	of a normal matrix, Linear Algebra and its Applications. 364(2003) 13–31.
	mondal2023degree S. Mondal, K. C. Das, Degree-Based Graph Entropy in Structure–Property Modeling, Entropy. 25(2023) 1092.
	nithya2023smallest P. Nithya, S. Elumalai, S. Balachandran, S. Mondal, Smallest abs index of
	unicyclic graphs with given girth, Journal of Applied Mathematics and Computing. 69(2023) 3675–3692.
	oboudi2019new M. R. Oboudi, A new lower bound for the energy of graphs, Linear Algebra and its
		Applications. 580(2019) 384–395.
	ramane2017status H. S. Ramane, A. S. Yalnaik, Status connectivity indices of graphs and its applications to the boiling point of benzenoid hydrocarbons, Journal of Applied
		Mathematics and Computing. 55 (2017) 609–627.
	zhou2010sum B. Zhou, N. Trinajstic, On sum-connectivity matrix and sum-connectivity energy of (molecular) graphs, Acta Chim. Slov. 57 (2010) 518–523.

 