# On spectrally optimal duals of frames generated by graphs

Deepshikha Department of Mathematics, Shyampur Siddheswari Mahavidyalaya,
University of Calcutta, West Bengal 711312, India. Email<EMAIL_ADDRESS>Aniruddha Samanta Theoretical Statistics and Mathematics Unit, Indian
Statistical Institute, Kolkata-700108, India. Email<EMAIL_ADDRESS>
###### Abstract

Recently, the concept of frames generated by graphs has been introduced in
[12]. In this paper, we study spectrally optimal dual frames of frames
generated by graphs. We show that if the frame is generated by a connected
graph, then its canonical dual frame is the unique spectrally optimal dual
frame for $1$-erasure and $2$-erasures. Further, we show that the canonical
dual frames of frames generated by disconnected graphs are non-unique
spectrally optimal dual frames for $1$-erasure and $2$-erasures.

AMS Subject Classification(2010): 05C40, 05C50, 42C15, 42C40, 46C05.

Keywords. Frames generated by graphs, Dual frames, Spectrally optimal dual
frames, Simple graphs, Laplacian matrix, Error operator.

##  1 Introduction

In order to solve certain complex problems in the non-harmonic Fourier series,
Duffin and Schaeffer [15] developed the idea of frames. Frames are redundant
sequences of vectors that span a vector space. The redundancy of frames makes
them very useful in many applications, including data and image processing,
signal transmission, image segmentation, coding theory, etc. Due to
redundancy, the data transmitted using frames can be reconstructed with
minimal error if erasures occur during the transmission.

Finite frames are widely used in frame theory due to their significant
relevance in applications. In [12], authors introduced a new class of finite
frames using simple graphs and called them frames generated by graphs. They
established some graph properties using frame properties and vice-versa.
Consequently, it provides a beautiful connection between the three classical
branches of mathematics, namely, frame theory, graph theory, and matrix
theory. Further, frames generated by graphs have some special properties, and
some of them are stated in the preliminaries section of this article.

The problem of erasures commonly occurs due to transmission losses or
disturbances during transmission, see [6, 7]. The problem of erasures can be
handled in two ways. One way is to find a frame that can minimize the maximal
error in reconstruction [17]. Another way is to find an optimal dual frame for
a pre-selected frame in order to minimize the maximal error of reconstruction,
see [18]. In this paper, we focus on the second approach. A dual frame of a
pre-selected frame that minimizes the maximal error of reconstruction is
called an optimal dual frame. Various measurements are used to measure the
error operator. If operator norm is used, then we use the term optimal dual
frame. If the spectral radius is used to measure the error operator, then dual
frames that minimize the maximum error are called spectrally optimal dual
frames (or, in short, $SOD$-frames). In [14], authors studied optimal dual
frames of frames generated by graphs. In this paper, we study the spectrally
optimal dual frame of frames generated by graphs. We establish some
interesting properties of $SOD$-frames for $1$-erasure and for $2$-erasures.

The outline of this paper is as follows. In Section 2, we provide the basic
definitions and results of frame theory, spectrally optimal dual frames,
spectral graph theory, and frames generated by graphs required in the rest of
the article. In Section 3, we study spectrally optimal dual frames of frames
generated by graphs for $1$-erasure. We show that the canonical dual frames of
frames generated by connected graphs are the unique spectrally optimal dual
frames for $1$-erasure. Further, if a frame is generated by a disconnected
graph, then its canonical dual is a non-unique $SOD$-frame for $1$-erasure.
Section 3 deals with spectrally optimal dual frames for $2$-erasures. First,
we compute the spectral radius of the error operator for $2$-erasures of
frames generated by graphs. We establish that canonical duals of frames
generated by graphs are $SOD$-frames for $2$-erasures. Further, we show that
they are non-unique $SOD$-frames for $2$-erasures if the frame is generated by
a disconnected graph.

##  2 Preliminaries

Throughout the rest of the article, we consider finite frames and
$\mathbb{C}^{k}$ is a $k$-dimensional Hilbert space. We use the notation $[n]$
to denote the set $\\{1,2,\ldots,n\\}$ and $M=[m_{i,j}]_{i\in[p],j\in[q]}$ to
denote the matrix $M$ of order $p\times q$ where $m_{i,j}$ is the $(i,j)$-th
entry of $M$. Also, $diag(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})$ denotes
the diagonal matrix of order $n$ with respective diagonal entries
$\alpha_{1},\alpha_{2},\ldots,\alpha_{n}$.

A sequence of vectors $\Phi:=\\{\phi_{i}\\}_{i\in[n]}$ in $\mathbb{C}^{k}$ is
called a _frame_ for $\mathbb{C}^{k}$ if there exist constants $0<A_{o}\leq
B_{o}$ such that for any $f\in\mathbb{C}^{k}$, we have

$\displaystyle A_{o}\|f\|^{2}\leq\sum_{i\in[n]}|\langle
f,\phi_{i}\rangle|^{2}\leq B_{o}\|f\|^{2}.$

The constants $A_{o}$ and $B_{o}$ are called the _lower_ and _upper frame
bounds_ of $\Phi$, respectively. For a frame $\Phi=\\{\phi_{i}\\}_{i\in[n]}$,
the _analysis operator_ $T_{\Phi}:\mathbb{C}^{k}\rightarrow\mathbb{C}^{n}$ is
defined as $T_{\Phi}(f)=\\{\langle f,\phi_{i}\rangle\\}_{i\in[n]}$. The
analysis operator $T_{\Phi}$ is linear and bounded. The adjoint of $T_{\Phi}$,
$T^{*}_{\Phi}:\mathbb{C}^{n}\rightarrow\mathbb{C}^{k}$, is called a _synthesis
operator_ and defined by
$T_{\Phi}(\\{\alpha_{i}\\}_{i\in[n]})=\sum\limits_{i\in[n]}\alpha_{i}\phi_{i}$.
The canonical matrix representation of the synthesis operator is a $k\times n$
matrix such that its $i^{\text{th}}$ column is the vector $\phi_{i}$ that is
$[T^{*}_{\Phi}]=[\phi_{1}\,\,\phi_{2}\,\,\cdots\,\,\phi_{n}]$. The _frame
operator_ $S_{\Phi}:\mathbb{C}^{k}\rightarrow\mathbb{C}^{k}$ is the
composition of the synthesis operator and analysis operator such that
$S_{\Phi}(f)=\sum\limits_{i\in[n]}\langle f,\phi_{i}\rangle\phi_{i}$. The
operator $S_{\Phi}$ is bounded, linear, self-adjoint, positive, and
invertible. The composition
$T_{\Phi}T_{\Phi}^{*}:\mathbb{C}^{n}\rightarrow\mathbb{C}^{n}$ is called the
_Gramian operator_ of $\Phi$. The canonical matrix representation of the
Gramian operator is said to be the _Gramian matrix_ , defined by
$\mathcal{G}_{\Phi}=[\langle\phi_{j},\phi_{i}\rangle]_{i,j\in[n]}$ where
$\langle\phi_{j},\phi_{i}\rangle$ is the $(i,j)$-th entry of
$\mathcal{G}_{\Phi}$. A frame $\Psi=\\{\psi_{i}\\}_{i\in[n]}$ of
$\mathbb{C}^{k}$ is called a _dual frame_ of $\Phi$ if for any
$f\in\mathbb{C}^{k}$, $f=\sum\limits_{i\in[n]}\langle
f,\phi_{i}\rangle\psi_{i}=\sum\limits_{i\in[n]}\langle
f,\psi_{i}\rangle\phi_{i}$. The sequence
$\\{S_{\Phi}^{-1}\phi_{i}\\}_{i\in[n]}$ is a also a dual frame of $\Phi$ and
it is called the _canonical dual frame_ of $\Phi$. We denote the set of all
dual frames of a frame $\Phi$ by $D_{\Phi}$. The dual frames which are not
canonical dual are called _alternate dual frames_. If
$\Psi=\\{\psi_{i}\\}_{i\in[n]}\in D_{\Phi}$ then there exist a sequence
$\\{\mu_{i}\\}_{i\in[n]}\subset\mathbb{C}^{k}$ such that
$\psi_{i}=S_{\Phi}^{-1}\phi_{i}+\mu_{i}$ for $i\in[n]$ and
$\sum\limits_{i\in[n]}\langle f,\mu_{i}\rangle\phi_{i}=0$ for all
$f\in\mathbb{C}^{k}$. We say two frames $\Phi_{1}=\\{\phi_{1,i}\\}_{i\in[n]}$
and $\Phi_{2}=\\{\phi_{2,i}\\}_{i\in[n]}$ are _unitary equivalent_ if there
exists a unitary operator $V:\mathbb{C}^{k}\rightarrow\mathbb{C}^{k}$ such
that $V(\phi_{1,i})=\psi_{1,i}$ for all $i\in[n]$. Refer to [4, 8, 9, 11] for
the general theory of frames and its applications.

Suppose $\Phi=\\{\phi_{i}\\}_{i\in[n]}$ is a frame for $\mathbb{C}^{k}$ and
$\Psi=\\{\psi_{i}\\}_{i\in[n]}$ is a dual frame of $\Phi$. Then for
$\Lambda\subset[n]$, the error operator
$E_{\Phi,\Psi,\Lambda}:\mathbb{C}^{k}\rightarrow\mathbb{C}^{k}$ is defined by

$\displaystyle E_{\Phi,\Psi,\Lambda}(f)=\sum_{i\in\Lambda}\langle
f,\phi_{i}\rangle\psi_{i}.$

For $1\leq r<n$, let
$\rho^{(r)}_{\Phi,\Psi}:=\max\\{\rho(E_{\Phi,\Psi,\Lambda}):\Lambda\subset[n]\text{
and }|\Lambda|=r\\}$, where $\rho(E_{\Phi,\Psi,\Lambda})$ is the spectral
radius of $E_{\Phi,\Psi,\Lambda}$. A dual frame $\Psi$ of the frame $\Phi$ is
called a _spectrally optimal dual frame_ (or simply _$SOD$ -frame_) for
$1$-erasure if
$\rho^{(1)}_{\Phi}:=\min\left\\{\rho^{(1)}_{\Phi,\widetilde{\Psi}}:\widetilde{\Psi}\in
D_{\Phi}\right\\}=\rho^{(1)}_{\Phi,\Psi}$. Spectrally optimal dual frames for
$r$-erasures are defined inductively. For $1<r<n$, a dual frame
$\Psi=\\{\psi_{i}\\}_{i\in[n]}$ of a frame $\Phi=\\{\phi_{i}\\}_{i\in[n]}$ is
called _spectrally optimal dual frame for $r$-erasures_ if $\Psi$ is an
spectrally optimal dual frame for $(r-1)$-erasures and
$\rho^{(r)}_{\Phi}:=\min\\{\rho^{(r)}_{\Phi,\widetilde{\Psi}}:\widetilde{\Psi}\in
D_{\Phi}\\}=\rho^{(r)}_{\Phi,\Psi}$. We denote the set of all $SOD$-frame of
$\Phi$ for $r$-erasures by $SOD_{\Phi}^{r}$.

######  Remark 2.1.

[22] If $\Phi=\\{\phi_{i}\\}_{i\in[n]}$ is a frame with dual frame
$\Psi=\\{\psi_{i}\\}_{i\in[n]}$, then
$\rho^{(1)}_{\Phi,\Psi}=\max\\{|\langle\psi_{i},\phi_{i}\rangle|:i\in[n]\\}$.

######  Theorem 2.1.

[22] Suppose $\Phi=\\{\phi_{i}\\}_{i\in[n]}$ is a frame for $\mathbb{C}^{k}$.
Then the following are equivalent:

  1. 1.

There exists a dual frame $\Psi=\\{\psi_{i}\\}_{i\in[n]}$ of $\Phi$ such that
$\rho^{(1)}_{\Phi,\Psi}=\rho^{(1)}_{\Phi}=\frac{k}{n}$.

  2. 2.

There exists a dual frame $\Psi=\\{\psi_{i}\\}_{i\in[n]}$ of $\Phi$ such that
$|\langle\psi_{i},\phi_{i}\rangle|=\frac{k}{n}$ for all $i\in[n]$.

######  Theorem 2.2.

[1] Suppose $\Phi=\\{\phi_{i}\\}_{i\in[n]}$ is a frame for $\mathbb{C}^{k}$
and $V:\mathbb{C}^{k}\rightarrow\mathbb{C}^{k}$ is an invertible operator.
Then a dual frame $\Psi=\\{\psi_{i}\\}_{i\in[n]}$ is (unique) spectrally
optimal dual frame of $\Phi$ for $r$-erasures if and only if $V\Psi$ is
(unique) spectrally optimal dual frame of $(V^{-1})^{*}\Phi$ for $r$-erasures.

For the detailed study of optimal dual frames and spectrally optimal dual
frames, readers may refer to [13, 19, 18, 22].

Graph theory is one of the classical branches of mathematics. Along with the
deep theoretical aspect of graph theory, it has many real-life applications,
which include measuring the connectivity of communication networks,
hierarchical clustering, ranking of hyperlinks in search engines, image
segmentation, and so on, see [21, 10, 5, 23]. Let $G=(V(G),E(G))$ be a simple
graph with the vertex set $V(G)=\\{v_{1},v_{2},\dots,v_{n}\\}$ and edge set
$E(G)$. The vertices $v_{i}$ and $v_{j}$ are said to be _adjacent_ if they are
connected by an edge and are denoted by $v_{i}\sim v_{j}$. The number of
vertices adjacent to $v_{i}$ is known as the _degree of the vertex_ $v_{i}$
and is denoted by $d(v_{i})$ or simply by $d_{i}$. The _degree matrix_ of a
graph $G$ on $n$ vertices, denoted by $D(G):=diag(d_{1},d_{2},\ldots,d_{n})$.
The _adjacency matrix_ of $G$ on $n$ vertices is $A(G):=(a_{i,j})_{n\times
n}$, where

$\displaystyle a_{i,j}=\begin{cases}1,\ \text{for }v_{i}\sim v_{j}\text{ and
}i\neq j\\\ 0,\text{ elsewhere.}\end{cases}$

The matrix $A(G)$ is simply written as $A$, if the graph $G$ is clear from the
context. The _Laplacian matrix_ of a graph $G$ is defined as
$L(G):=D(G)-A(G)$. We simply write $L(G)$ as $L$ if $G$ is clear from the
context. The Laplacian matrix is a positive semi-definite matrix. For a simple
graph $G$ on $n$ vertices with $k$ components, the rank of $L(G)$ is $n-k$.
For a detailed study of matrices associated with simple graphs and other
classes of graphs, readers may see [2, 24, 20, 25, 16, 3] and references
therein. The transpose of a matrix $M$ is denoted by $M^{t}$. If $M_{1}$ and
$M_{2}$ are matrices, then $M_{1}\oplus M_{2}$ denotes the block matrix
$\left[\begin{array}[]{cc}M_{1}&\boldsymbol{0}\\\
\boldsymbol{0}&M_{2}\end{array}\right]$.

In [12], authors used Laplacian matrices of graphs to generate finite frames.
First, let us see the definition of $L_{G}(n,k)$-frames.

######  Definition 2.1.

Let $G$ be a simple graph on $n$ vertices with $n-k$ components. Suppose $L$
is the Laplacian matrix of $G$ such that
$L=M\,diag(\lambda_{1},\lambda_{2},\ldots,\lambda_{k},0,\ldots,0)\,M^{*}$. If
$\\{e_{i}\\}_{i\in[n]}$ is the standard canonical orthonormal basis of
$\mathbb{C}^{n}$ and
$B=diag\left(\sqrt{\lambda_{1}},\ldots,\sqrt{\lambda_{k}}\right)M_{1}^{*}$,
where $M_{1}$ is a submatrix of $M$ formed by the first $k$ columns then
$\\{B(e_{i})\\}_{i\in[n]}$ is a frame for $\mathbb{C}^{k}$ and it is called an
$L_{G}(n,k)$-frame for $\mathbb{C}^{k}$.

Next, we have the definition of frames generated by graphs.

######  Definition 2.2.

Let $\Phi=\\{\phi_{i}\\}_{i\in[n]}$ be a frame for $\mathbb{C}^{k}$ with
Gramian matrix $\mathcal{G}_{\Phi}$. If $G$ is a graph with Laplacian matrix
$L$ such that $\mathcal{G}_{\Phi}=L$, then $\Phi$ is called a frame generated
by the graph $G$ for $\mathbb{C}^{k}$. In short, we call $\Phi$ as a
$G(n,k)$-frame for $\mathbb{C}^{k}$.

In [12], the authors have shown that every $L_{G}(n,k)$-frame is a
$G(n,k)$-frame. Let us see some results related to frames generated by graphs.

######  Theorem 2.3 ([12]).

If $G$ is a graph, then frames generated by $G$ are unitary equivalent.

The next theorem shows that the frame operator of any $L_{G}(n,k)$-frame is a
diagonal matrix.

######  Theorem 2.4 ([12]).

If $G$ is a graph and $\Phi$ is an $L_{G}(n,k)$-frame then the frame operator
$S_{\Phi}$ of the frame $\Phi$ is a diagonal matrix. Further, if
$\lambda_{1},\lambda_{2},\ldots,\lambda_{k}$ are the non-zero eigenvalues of
the Laplacian matrix of $G$ then
$S_{\Phi}=diag(\lambda_{1},\lambda_{2},\ldots,\lambda_{k})$.

The following theorem gives the family of dual frames of the frames generated
by graphs.

######  Theorem 2.5.

Let $G$ be a simple graph with components $G_{1},G_{2},\ldots,G_{m}$ having
vertex sets
$\\{v_{1},v_{2},\ldots,v_{n_{1}}\\},\\{v_{n_{1}+1},v_{n_{1}+2},\ldots,v_{n_{2}}\\},\ldots,\\{v_{n_{m-1}+1},v_{n_{m-1}+2},\ldots,v_{n_{m}}(=v_{n})\\}$,
respectively. If $\Phi=\\{\phi_{i}\\}_{i\in[n]}$ is a $G(n,n-m)$-frame with
the frame operator $S_{\Phi}$, then any dual frame of $\Phi$ is of the form
$\\{S_{\Phi}^{-1}\phi_{i}+\nu_{1}\\}_{i=1}^{n_{1}}\bigcup\\{S_{\Phi}^{-1}\phi_{i}+\nu_{2}\\}_{i=n_{1}+1}^{n_{2}}\bigcup\cdots\bigcup\\{S_{\Phi}^{-1}\phi_{i}+\nu_{m}\\}_{i=n_{m-1}+1}^{n}$
where $\nu_{1},\nu_{2},\ldots,\nu_{m}$ are arbitrary vectors in
$\mathbb{C}^{n-m}$.

##  3 On spectrally optimal dual frame for $1$-erasure

We begin this article by showing that if the canonical dual frame of a frame
generated by a graph $G$ is a spectrally optimal dual frame for $r$-erasures,
then the canonical dual of any frame generated by $G$ is a spectrally optimal
dual frame for $r$-erasures.

######  Theorem 3.1.

Let $G$ be any graph with $n$ vertices. If
$\Phi_{1}=\\{\phi_{1,i}\\}_{i\in[n]}$ and
$\Phi_{2}=\\{\phi_{2,i}\\}_{i\in[n]}$ are $G(n,k)$-frames for
$\mathbb{C}^{k}$, then the canonical dual of $\Phi_{1}$ is (unique) spectrally
optimal dual frame of $\Phi_{1}$ for $r$-erasures if and only if the canonical
dual of $\Phi_{2}$ is (unique) spectrally optimal dual frame of $\Phi_{2}$ for
$r$-erasures.

###### Proof.

Since $\Phi_{1}$ and $\Phi_{2}$ are $G(n,k)$-frames, by Theorem 2.3, there is
a unitary operator $V:\mathbb{C}^{k}\rightarrow\mathbb{C}^{k}$ such that
$V(\phi_{1,i})=\phi_{2,i}$ for all $i\in[n]$. Let $S_{\Phi_{1}}$ and
$S_{\Phi_{2}}$ are the frame operators of $\Phi_{1}$ and $\Phi_{2}$,
respectively. Then, for any $f\in\mathbb{C}^{k}$, we have

$\displaystyle S_{\Phi_{2}}(f)$ $\displaystyle=\sum\limits_{i\in[n]}\langle
f,\phi_{2,i}\rangle\phi_{2,i}$ $\displaystyle=\sum\limits_{i\in[n]}\langle
f,V\phi_{1,i}\rangle V\phi_{1,i}$ $\displaystyle=VS_{\Phi_{1}}V^{*}(f).$

Thus, $S_{\Phi_{2}}=VS_{\Phi_{1}}V^{*}$ that is
$S_{\Phi_{1}}^{-1}=V^{*}S_{\Phi_{2}}^{-1}V$. Then, by Theorem 2.2,
$S_{\Phi_{1}}^{-1}\Phi_{1}$ is (unique) $SOD$-frame of $\Phi_{1}$ for
$r$-erasures if and only if
$VS_{\Phi_{1}}^{-1}\Phi_{1}=VV^{*}S_{\Phi_{2}}^{-1}VV^{*}\Phi_{2}=S_{\Phi_{2}}^{-1}\Phi_{2}$
is (unique) $SOD$-frame of $(V^{-1})^{*}\Phi_{1}=V\Phi_{1}=\Phi_{2}$ for
$r$-erasures. ∎

In the following theorem, we show that the canonical dual of a frame generated
by the connected graph is the unique $SOD$-frame for $1$-erasure.

######  Theorem 3.2.

Suppose $G$ is a connected graph with $n$ vertices. If
$\Phi=\\{\phi_{i}\\}_{i\in[n]}$ is an $L_{G}(n,n-1)$-frame for
$\mathbb{C}^{n-1}$, then canonical dual frame of $\Phi$ is the unique
spectrally optimal dual frame of $\Phi$ for $1$-erasure. Further,
$\rho^{(1)}_{\Phi}=\frac{n-1}{n}$.

###### Proof.

Let $L$ be the Laplacian matrix of $G$ with eigenvalues
$\lambda_{1},\lambda_{2},\ldots,\lambda_{n-1},0$. Then there exist an
orthogonal matrix $M$ consisting of eigenvectors of $L$ and
$D=diag(\lambda_{1},\lambda_{2},\ldots,\lambda_{n-1},0)$ such that $L=MDM^{*}$
and $\phi_{i}=D_{1}M_{1}^{*}(e_{i})$ where
$D_{1}=diag(\sqrt{\lambda_{1}},\sqrt{\lambda_{2}},\ldots,\sqrt{\lambda_{n-1}})$,
$M_{1}$ is the matrix obtained from $M$ by taking the first $n-1$ columns of
$M$ and $\\{e_{i}\\}_{i\in[n]}$ is the standard canonical orthonormal basis of
$\mathbb{C}^{n}$. Note that $[1\,1\,\cdots\,1]^{t}$ is an eigenvector of $L$
corresponding to the eigenvalue $0$. Hence, the last column of the matrix $M$
is
$\left[\frac{1}{\sqrt{n}}\,\frac{1}{\sqrt{n}}\,\cdots\,\frac{1}{\sqrt{n}}\right]^{t}$.
Suppose $S_{\Phi}$ is the frame operator of $\Phi$. Then, by Theorem 2.4,
$S_{\Phi}=diag(\lambda_{1},\lambda_{2},\ldots,\lambda_{n-1})$ and hence
$S_{\Phi}^{1/2}=diag(\sqrt{\lambda_{1}},\sqrt{\lambda_{2}},\ldots,\sqrt{\lambda_{n-1}})=D_{1}$.
Thus $S_{\Phi}^{-1/2}=D_{1}^{-1}$. Then we have

$\displaystyle|\langle
S_{\Phi}^{-1}\phi_{i},\phi_{i}\rangle|=\|S_{\Phi}^{-1/2}\phi_{i}\|^{2}=\|D_{1}^{-1}D_{1}M_{1}^{*}(e_{i})\|^{2}=\|M_{1}^{*}(e_{i})\|^{2}=1-\frac{1}{n}.$

Thus, $|\langle S_{\Phi}^{-1}\phi_{i},\phi_{i}\rangle|=\frac{n-1}{n}$ for all
$i\in[n]$. Hence, by Theorem 2.1,
$\rho_{\Phi,S_{\Phi}^{-1}\Phi}^{(1)}=\rho_{\Phi}^{(1)}=\frac{n-1}{n}$.
Therefore, the canonical dual $S_{\Phi}^{-1}\Phi$ is an $SOD$-frame of $\Phi$
for $1$-erasure. Let $\Psi=\\{\psi_{i}\\}_{i\in[n]}$ be a spectrally optimal
dual frame of $\Phi$ for $1$-erasure. Then, by Theorem 2.5, there exists a
vector $\mu\in\mathbb{C}^{n-1}$ such that $\psi_{i}=S_{\Phi}^{-1}\phi_{i}+\mu$
for all $i\in[n]$. Then $\sum\limits_{i\in[n]}\langle f,\mu\rangle\phi_{i}=0$
for all $f\in\mathbb{C}^{n-1}$. Thus,
$\sum\limits_{i\in[n]}\langle\mu,\phi_{i}\rangle=0$ and we have

$\displaystyle\rho_{\Phi,\Psi}^{(1)}$ $\displaystyle=\max\\{|\langle
S_{\Phi}^{-1}\phi_{i}+\mu,\phi_{i}\rangle|:i\in[n]\\}$
$\displaystyle=\max\\{|\|S_{\Phi}^{-1/2}\phi_{i}\|^{2}+\langle\mu,\phi_{i}\rangle|:i\in[n]\\}$
$\displaystyle=\max\left\\{\left|\frac{n-1}{n}+\langle\mu,\phi_{i}\rangle\right|:i\in[n]\right\\}$
$\displaystyle=\max\left\\{\sqrt{\left|\frac{n-1}{n}+Re\langle\mu,\phi_{i}\rangle\right|^{2}+|Im\langle\mu,\phi_{i}\rangle|^{2}}:i\in[n]\right\\}$
$\displaystyle=\frac{n-1}{n}.$

Thus,
$\left|\frac{n-1}{n}+Re\langle\mu,\phi_{i}\rangle\right|\leq\frac{n-1}{n}$ for
all $i\in[n]$. Hence, $Re\langle\mu,\phi_{i}\rangle\leq 0$ for all $i\in[n]$.
Since $\sum\limits_{i\in[n]}\langle\mu,\phi_{i}\rangle=0$ ,
$Re\langle\mu,\phi_{i}\rangle=0$ for $i\in[n]$. Thus,
$\rho_{\Phi,\Psi}^{(1)}=\max\left\\{\sqrt{\left|\frac{n-1}{n}\right|^{2}+|Im\langle\mu,\phi_{i}\rangle|^{2}}:i\in[n]\right\\}\break=\frac{n-1}{n}$.
This gives that $|Im\langle\mu,\phi_{i}\rangle|^{2}=0$ for all $i\in[n]$.
Hence, $Im\langle\mu,\phi_{i}\rangle=0$ for all $i\in[n]$. Then
$\langle\mu,\phi_{i}\rangle=0$ for all $i\in[n]$ and hence $\mu=0$. Thus,
$\Psi=S_{\Phi}^{-1}\Phi$ and $S_{\Phi}^{-1}\Phi$ is the unique spectrally
optimal dual frame of $\Phi$ for $1$-erasure. ∎

######  Corollary 3.1.

Suppose $G$ is a connected graph with $n$ vertices. If
$\Phi=\\{\phi_{i}\\}_{i\in[n]}$ is a $G(n,n-1)$-frame for $\mathbb{C}^{n-1}$,
then canonical dual frame of $\Phi$ is the unique spectrally optimal dual
frame of $\Phi$ for $1$-erasure.

###### Proof.

Proof follows from Theorem 3.1 and Theorem 3.2. ∎

In Theorem 3.2, we consider $G$ to be a connected graph. In the following
theorem, we show that Theorem 3.2 is partially true for disconnected graphs.

######  Theorem 3.3.

Suppose $G$ is a graph with $n$ vertices and $k>1$ connected components. If
$\Phi=\\{\phi_{i}\\}_{i\in[n]}$ is an $L_{G}(n,n-k)$-frame for
$\mathbb{C}^{n-k}$, then canonical dual frame of $\Phi$ is a non-unique
spectrally optimal dual frame of $\Phi$ for $1$-erasure.

###### Proof.

Let $G_{1},G_{2},\ldots,G_{k}$ be the connected components of $G$ with
$|V(G_{i})|=n_{i}$ for $i\in[k]$. Let the vertex set of $G_{i}$ be
$\\{v_{l_{i-1}+1},v_{l_{i-1}+2},\ldots,v_{l_{i}}\\}$ for $i\in[k]$ where
$l_{0}=0$ and for $1\leq i\leq k$, $l_{i}=n_{1}+n_{2}+\cdots+n_{i}$. If
$L_{i}$ is the Laplacian matrix of $G_{i}$, then the Laplacian matrix of $G$
is $L=L_{1}\oplus L_{2}\oplus\cdots\oplus L_{k}$. Suppose the the spectral
decomposition of $L_{i}$ is $M_{i}D_{i}M_{i}^{*}$w that is
$L_{i}=M_{i}D_{i}M_{i}^{*}$ where
$D_{i}=diag(\lambda_{1}^{i},\ldots,\lambda_{n_{i}-1}^{i},0)$ and $M_{i}$ is an
orthogonal matrix consist of eigenvectors of $L_{i}$. Then $L=MDM^{*}$ where
$M=M_{1}\oplus\cdots\oplus M_{k}$ and $D=D_{1}\oplus\cdots\oplus D_{k}$. Let
$\widetilde{D_{i}}=diag\left(\sqrt{\lambda_{1}^{i}},\ldots,\sqrt{\lambda_{n_{i}-1}^{i}}\right)$
and $\widetilde{M_{i}}$ be obtained from $M_{i}$ by taking the first $n_{i}-1$
columns for $i\in[k]$. Then $\widetilde{D_{i}}\widetilde{M_{i}}^{*}$ is the
synthesis operator of frame say $\Phi_{i}=\\{\phi_{i,j}\\}_{j\in[n_{i}]}$ for
$\mathbb{C}^{n_{i}-1}$. Let $S_{\Phi_{i}}$ be the frame operator of $\Phi_{i}$
for $i\in[k]$.

Let $\widetilde{D}=\widetilde{D_{1}}\oplus\cdots\oplus\widetilde{D_{k}}$ and
$\widetilde{M}=\widetilde{M_{1}}\oplus\cdots\oplus\widetilde{M_{k}}$. Then
$\widetilde{\Phi}=\\{\widetilde{\phi}_{i}\\}_{i\in[n]}=\\{\widetilde{D}\widetilde{M}^{*}(e_{i})\\}_{i\in[n]}$
is an $L_{G}(n,n-k)$-frame for $\mathbb{C}^{n-k}$. Also, the synthesis
operator $T_{\widetilde{\Phi}}^{*}$ of $\widetilde{\Phi}$ is

$\displaystyle[T_{\widetilde{\Phi}}^{*}]$
$\displaystyle=[T_{\Phi_{1}}^{*}]\oplus[T_{\Phi_{2}}^{*}]\oplus\cdots\oplus[T_{\Phi_{k}}^{*}]$
$\displaystyle=\left[\begin{array}[]{cccccccccc}\phi_{1,1}&\cdots&\phi_{1,n_{1}}&0&\cdots&0&\cdots&0&\cdots&0\\\
0&\cdots&0&\phi_{2,1}&\cdots&\phi_{2,n_{2}}&\cdots&0&\cdots&0\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\ddots&\vdots&\ddots&\vdots\\\
0&\cdots&0&0&\cdots&0&\cdots&\phi_{k,1}&\cdots&\phi_{k,n_{k}}\end{array}\right]$
$\displaystyle=\\{\widetilde{\phi}_{i}\\}_{i\in[n]}$

By Theorem 3.1, it is enough to prove that the canonical dual of
$\widetilde{\Phi}$ is a non-unique spectrally optimal dual frame of
$\widetilde{\Phi}$ for $1$-erasure.

Let $S_{\widetilde{\Phi}}$ be the frame operator of $\widetilde{\Phi}$. Then
$S_{\widetilde{\Phi}}=S_{\Phi_{1}}\oplus\cdots\oplus S_{\Phi_{k}}$. Suppose
$\max\\{|\langle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{i},\widetilde{\phi}_{i}\rangle|:i\in[n]\\}=|\langle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{j},\widetilde{\phi}_{j}\rangle|$
where $j\in[l_{r-1}+1,\ldots,l_{r}]$ for some $r\in[k]$. Then
$\rho^{(1)}_{\widetilde{\Phi},S_{\widetilde{\Phi}}^{-1}\widetilde{\Phi}}=|\langle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{j},\widetilde{\phi}_{j}\rangle|=1-\frac{1}{n_{r}}=\frac{n_{r}-1}{n_{r}}$.
By Theorem 3.2,
$\rho^{(1)}_{\Phi_{r}}=\rho^{(1)}_{\Phi_{r},S_{\Phi_{r}}^{-1}\Phi_{r}}=\frac{n_{r}-1}{n_{r}}$.

Suppose $\Psi=\\{\psi_{i}\\}_{i\in[n]}$ is a dual frame of $\widetilde{\Phi}$.
Since $\Psi$ is a dual frame, there exist vectors
$\mu_{1},\mu_{2},\ldots,\mu_{k}$ in $\mathbb{C}^{n-k}$ such that
$\psi_{i}=S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{i}+\mu_{j}$ for
$i\in\\{l_{j-1}+1,l_{j-1}+2,\ldots,l_{j}\\}$. Let
$\mu_{j}=\left[\begin{array}[]{c}\nu_{1}\\\ \nu_{2}\\\ \vdots\\\
\nu_{k}\end{array}\right]$ where $\nu_{i}\in\mathbb{C}^{n_{i}-1}$. Then by
Theorem 2.5,
$\Psi_{r}=\\{S_{\Phi_{r}}^{-1}\phi_{r,i}+\nu_{r}\\}_{i\in[n_{r}]}$ is a dual
frame of $\Phi_{r}$. By Theorem 3.2,
$\rho_{\Phi_{r},\Psi_{r}}^{(1)}\geq\frac{n_{r}-1}{n_{r}}$. Then

$\displaystyle\rho^{(1)}_{\widetilde{\Phi},\Psi}$
$\displaystyle=\max\\{|\langle\psi_{i},\widetilde{\phi}_{i}\rangle|:i\in[n]\\}$
$\displaystyle\geq\max\\{|\langle\psi_{i},\widetilde{\phi}_{i}\rangle|:i\in\\{l_{r-1}+1,\ldots,l_{r}\\}\\}$
$\displaystyle=\max\\{|\langle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{i}+\mu_{j},\widetilde{\phi}_{i}\rangle|:i\in\\{l_{r-1}+1,\ldots,l_{r}\\}\\}$
$\displaystyle=\max\\{|\langle
S_{\Phi_{r}}^{-1}\widetilde{\phi}_{r,i}+\nu_{r},\widetilde{\phi}_{r,i}\rangle|:i\in[n_{r}]\\}$
$\displaystyle=\rho^{(1)}_{\Phi_{r},\Psi_{r}}$
$\displaystyle\geq\frac{n_{r}-1}{n_{r}}$
$\displaystyle=\rho^{(1)}_{\widetilde{\Phi},S_{\widetilde{\Phi}}^{-1}\widetilde{\Phi}}=\rho^{(1)}_{\widetilde{\Phi}}.$

Therefore, $S_{\widetilde{\Phi}}^{-1}\widetilde{\Phi}$ is an $SOD$-frame of
$\widetilde{\Phi}$ for $1$-erasure. Then by Theorem 3.1, $S_{\Phi}^{-1}\Phi$
is an $SOD$-frame of $\Phi$ for $1$-erasure.

Let $\mu=[\mu_{i}]_{i\in[n]}\in\mathbb{C}^{n-k}$ such that
$\mu_{i}=\begin{cases}0,\text{ if }i\in[l_{1}]\\\ 1,\text{ otherwise}\\\
\end{cases}$. By Theorem 2.5,
$\Psi_{1}=\\{\psi_{1,i}\\}_{i\in[n]}=\\{S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{i}+\mu\\}_{i\in[l_{1}]}\cup\\{S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{i}\\}_{i\in[n]\setminus[l_{1}]}$
is a dual frame of $\widetilde{\Phi}$. Note that $\Psi_{1}$ is not the
canonical dual frame of $\widetilde{\Phi}$. We have

$\displaystyle\rho_{\widetilde{\Phi},\Psi_{1}}^{(1)}$
$\displaystyle=\max\\{|\langle\psi_{1,i},\widetilde{\phi}_{i}\rangle|:i\in[n]\\}$
$\displaystyle=\max(\\{|\langle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{i}+\mu,\widetilde{\phi}_{i}\rangle|:i\in[l_{1}]\\}\cup\\{|\langle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{i},\widetilde{\phi}_{i}\rangle|:i\in[n]\setminus[l_{1}]\\})$
$\displaystyle=\max(\\{|\langle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{i},\widetilde{\phi}_{i}\rangle+\langle\mu,\widetilde{\phi}_{i}\rangle|:i\in[l_{1}]\\}\cup\\{|\langle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{i},\widetilde{\phi}_{i}\rangle|:i\in[n]\setminus[l_{1}]\\})$
$\displaystyle=\max(\\{|\langle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{i},\widetilde{\phi}_{i}\rangle:i\in[n]\\}\cup\\{|\langle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{i},\widetilde{\phi}_{i}\rangle|:i\in[n]\\})$
$\displaystyle=\rho_{\widetilde{\Phi},S_{\widetilde{\Phi}}^{-1}\widetilde{\Phi}}^{(1)}.$

Hence, $\Psi_{1}$ is an $SOD$-frame of $\widetilde{\Phi}$ for $1$-erasure.
Therefore, by Theorem 3.1, the canonical dual of $\Phi$ is a non-unique
$SOD$-frame of $\Phi$ for $1$-erasure. ∎

######  Corollary 3.2.

Suppose $G$ is a graph with $n$ vertices and $k>1$ connected components. If
$\Phi=\\{\phi_{i}\\}_{i\in[n]}$ is a $G(n,n-k)$-frame for $\mathbb{C}^{n-k}$,
then the canonical dual frame of $\Phi$ is a non-unique spectrally optimal
dual frame of $\Phi$ for $1$-erasure.

In the following example, we show explicitly that the canonical dual frames of
frames generated by disconnected graphs are non-unique spectrally optimal dual
frames for $1$-erasure.

######  Example 3.1.

Consider the graph $G$ given in Figure 1.

Figure 1:  Graph $G$

The graph $G$ is a disconnected graph. The Laplacian matrix of $G$ is
$L(G)=\left[\begin{array}[]{ccccc}2&-1&-1&0&0\\\ -1&2&-1&0&0\\\ -1&-1&2&0&0\\\
0&0&0&1&-1\\\ 0&0&0&-1&1\end{array}\right]$.

Consider the frame
$\Phi=\\{\phi_{i}\\}_{i\in[5]}=\left\\{\left[\begin{array}[]{c}\frac{\sqrt{3}}{\sqrt{2}}\\\\[5.69054pt]
\frac{-1}{\sqrt{2}}\\\\[5.69054pt]
0\end{array}\right],\left[\begin{array}[]{c}0\\\\[5.69054pt]
\sqrt{2}\\\\[5.69054pt]
0\end{array}\right],\left[\begin{array}[]{c}\frac{-\sqrt{3}}{\sqrt{2}}\\\\[5.69054pt]
\frac{-1}{\sqrt{2}}\\\\[5.69054pt]
0\end{array}\right],\left[\begin{array}[]{c}0\\\\[5.69054pt] 0\\\\[5.69054pt]
1\end{array}\right],\left[\begin{array}[]{c}0\\\\[5.69054pt] 0\\\\[5.69054pt]
-1\end{array}\right]\right\\}$ of $\mathbb{C}^{3}$.
Then the Gramian matrix of $\Phi$ is
$\mathcal{G}_{\Phi}=\left[\begin{array}[]{ccccc}2&-1&-1&0&0\\\ -1&2&-1&0&0\\\
-1&-1&2&0&0\\\ 0&0&0&1&-1\\\ 0&0&0&-1&1\end{array}\right]=L(G)$. Thus, $\Phi$
is a
$G(5,3)$-frame for $\mathbb{C}^{3}$. Let $S_{\Phi}$ be the frame operator of
$\Phi$. Then the canonical dual frame of
$\Phi$ is
$S_{\Phi}^{-1}\Phi=\left\\{\left[\begin{array}[]{c}\frac{1}{\sqrt{6}}\\\\[5.69054pt]
\frac{-1}{3\sqrt{2}}\\\\[5.69054pt]
0\end{array}\right],\left[\begin{array}[]{c}0\\\\[5.69054pt]
\frac{2}{3\sqrt{2}}\\\\[5.69054pt]
0\end{array}\right],\left[\begin{array}[]{c}\frac{-1}{\sqrt{6}}\\\\[5.69054pt]
\frac{-1}{3\sqrt{2}}\\\\[5.69054pt]
0\end{array}\right],\left[\begin{array}[]{c}0\\\\[5.69054pt] 0\\\\[5.69054pt]
\frac{1}{2}\end{array}\right],\left[\begin{array}[]{c}0\\\\[5.69054pt]
0\\\\[5.69054pt] \frac{-1}{2}\end{array}\right]\right\\}$. By Theorem 2.5, the
set of
dual frames of $\Phi$ is

$D_{\Phi}=\left\\{\left\\{\left[\begin{array}[]{c}\frac{1}{\sqrt{6}}+a\\\\[5.69054pt]
\frac{-1}{3\sqrt{2}}+b\\\\[5.69054pt]
c\end{array}\right],\left[\begin{array}[]{c}a\\\\[5.69054pt]
\frac{2}{3\sqrt{2}}+b\\\\[5.69054pt]
c\end{array}\right],\left[\begin{array}[]{c}\frac{-1}{\sqrt{6}}+a\\\\[5.69054pt]
\frac{-1}{3\sqrt{2}}+b\\\\[5.69054pt]
c\end{array}\right],\left[\begin{array}[]{c}e\\\\[5.69054pt] f\\\\[5.69054pt]
\frac{1}{2}+g\end{array}\right],\left[\begin{array}[]{c}e\\\\[5.69054pt]
f\\\\[5.69054pt]
\frac{-1}{2}+g\end{array}\right]\right\\}:a,b,c,e,f,g\in\mathbb{C}\right\\}$

.

Consider $\rho^{(1)}_{\Phi,S_{\Phi}^{-1}\Phi}=\max\\{|\langle
S_{\Phi}^{-1}\phi_{i},\phi_{i}\rangle|:i\in[5]\\}=\max\left\\{\frac{2}{3},\frac{1}{2}\right\\}=\frac{2}{3}$.
Let $\Psi=\\{\psi_{i}\\}_{i\in[5]}$ be any dual frame of $\Phi$. Then there
exist scalars $a_{o},b_{o},c_{o},d_{o},e_{o},f_{o},g_{o}$ such that
$\Psi=\\{\psi_{i}\\}_{i\in[5]}=\left\\{\left[\begin{array}[]{c}\frac{1}{\sqrt{6}}+a_{o}\\\\[2.84526pt]
\frac{-1}{3\sqrt{2}}+b_{o}\\\\[2.84526pt]
c_{o}\end{array}\right],\left[\begin{array}[]{c}a_{o}\\\\[2.84526pt]
\frac{2}{3\sqrt{2}}+b_{o}\\\\[2.84526pt]
c_{o}\end{array}\right],\left[\begin{array}[]{c}\frac{-1}{\sqrt{6}}+a_{o}\\\\[2.84526pt]
\frac{-1}{3\sqrt{2}}+b_{o}\\\\[2.84526pt]
c_{o}\end{array}\right],\left[\begin{array}[]{c}e_{o}\\\\[2.84526pt]
f_{o}\\\\[2.84526pt]
\frac{1}{2}+g_{o}\end{array}\right],\left[\begin{array}[]{c}e_{o}\\\\[2.84526pt]
f_{o}\\\\[2.84526pt] \frac{-1}{2}+g_{o}\end{array}\right]\right\\}$. Then

$\displaystyle\rho^{(1)}_{\Phi,\Psi}$
$\displaystyle=\max\\{|\langle\psi_{i},\phi_{i}\rangle|:i\in[5]\\}$
$\displaystyle=\max\left\\{\frac{2}{3}+\frac{\sqrt{3}}{\sqrt{2}}a_{o}-\frac{1}{\sqrt{2}}b_{o},\frac{2}{3}+\sqrt{2}b_{o},\frac{2}{3}-\frac{\sqrt{3}}{\sqrt{2}}a_{o}-\frac{1}{\sqrt{2}}b_{o},\frac{1}{2}+g_{o},\frac{1}{2}-g_{o}\right\\}$
$\displaystyle\geq\frac{2}{3}=\rho^{(1)}_{\Phi,S_{\Phi}^{-1}\Phi}.$

Thus, $S^{-1}\Phi$ is an $SOD$-frame of $\Phi$ for $1$-erasure. Consider

$\Psi_{1}=\\{\psi_{1,i}\\}_{i\in[5]}=\left\\{\left[\begin{array}[]{c}\frac{1}{\sqrt{6}}\\\\[2.84526pt]
\frac{-1}{3\sqrt{2}}\\\\[2.84526pt]
1\end{array}\right],\left[\begin{array}[]{c}0\\\\[2.84526pt]
\frac{2}{3\sqrt{2}}\\\\[2.84526pt]
1\end{array}\right],\left[\begin{array}[]{c}\frac{-1}{\sqrt{6}}\\\\[2.84526pt]
\frac{-1}{3\sqrt{2}}\\\\[2.84526pt]
1\end{array}\right],\left[\begin{array}[]{c}0\\\\[2.84526pt] 0\\\\[2.84526pt]
\frac{1}{2}\end{array}\right],\left[\begin{array}[]{c}0\\\\[2.84526pt]
0\\\\[2.84526pt] \frac{-1}{2}\end{array}\right]\right\\}.$

Then $\Psi_{1}$ is a dual frame of $\Phi$. Also,
$\rho^{(1)}_{\Psi,\Phi}=\max\\{|\langle\psi_{1,i},\phi_{i}\rangle|:i\in[5]\\}=\frac{2}{3}$.
Thus, $\Psi_{1}$ is an alternate dual frame of $\Phi$ and an $SOD$-frame of
$\Phi$ for $1$-erasure. Thus, the canonical dual of $\Phi$ is a non-unique
spectrally optimal dual of $\Phi$ for $1$-erasure.

##  4 On Spectrally optimal dual frames for $2$-erasures

In this section, we study spectrally optimal dual frames of frames generated
by graphs for $2$-erasures. In the previous section, we establish that frames
generated by graphs are spectrally optimal dual frames for $1$-erasure. Here,
we begin by measuring the error operator if canonical dual is used and
$2$-erasures occur.

######  Theorem 4.1.

Suppose $G$ is a connected graph on $n$ vertices and
$\Phi=\\{\phi_{i}\\}_{i\in[n]}$ is an $L_{G}(n,n-1)$-frame for
$\mathbb{C}^{n-1}$. If $S_{\Phi}$ is the frame operator of $\Phi$ then
$\rho_{\Phi,S_{\Phi}^{-1}\Phi}^{(2)}=1$.

###### Proof.

Let $L$ be the Laplacian matrix of $G$. Since $\Phi$ is an
$L_{G}(n,n-1)$-frame, there exist an orthogonal matrix $M$ and diagonal matrix
$D=diag(\lambda_{1},\lambda_{2},\ldots,\lambda_{n-1},0)$ such that $L=MDM^{*}$
and $\Phi=D_{1}M_{1}^{*}$ where $M_{1}$ is formed by taking the first $n-1$
columns of $M$ and
$D_{1}=diag(\sqrt{\lambda_{1}},\sqrt{\lambda_{2}},\ldots,\sqrt{\lambda_{n-1}})$.
Note that the last column of $M$ is
$\left[\begin{array}[]{c}\frac{1}{\sqrt{n}}\\\\[2.84526pt]
\frac{1}{\sqrt{n}}\\\\[2.84526pt] \vdots\\\ \frac{1}{\sqrt{n}}\\\\[2.84526pt]
\end{array}\right]$. Then, by Theorem 2.4,
$S_{\Phi}=diag(\lambda_{1},\ldots,\lambda_{n-1})$. Let
$\Lambda=\\{i,j\\}\subset[n]$ be arbitrary. Then, the error operator
$E_{\Phi,S_{\Phi}^{-1}\Phi,\Lambda}:\mathbb{C}^{k}\rightarrow\mathbb{C}^{k}$
is defined by

$E_{\Phi,S_{\Phi}^{-1}\Phi,\Lambda}(f)=\langle f,\phi_{i}\rangle
S_{\Phi}^{-1}\phi_{i}+\langle f,\phi_{j}\rangle S_{\Phi}^{-1}\phi_{j}.$

Consider

$\displaystyle
E_{\Phi,S_{\Phi}^{-1}\Phi,\Lambda}(S_{\Phi}^{-1}\phi_{i}+S_{\Phi}^{-1}\phi_{j})$
$\displaystyle=\langle
S_{\Phi}^{-1}\phi_{i}+S_{\Phi}^{-1}\phi_{j},\phi_{i}\rangle
S_{\Phi}^{-1}\phi_{i}+\langle
S_{\Phi}^{-1}\phi_{i}+S_{\Phi}^{-1}\phi_{j},\phi_{j}\rangle
S_{\Phi}^{-1}\phi_{j}$ $\displaystyle=\langle
S_{\Phi}^{-1}\phi_{i},\phi_{i}\rangle S_{\Phi}^{-1}\phi_{i}+\langle
S_{\Phi}^{-1}\phi_{j},\phi_{i}\rangle S_{\Phi}^{-1}\phi_{i}+\langle
S_{\Phi}^{-1}\phi_{i},\phi_{j}\rangle S_{\Phi}^{-1}\phi_{j}$
$\displaystyle\qquad\qquad\qquad\qquad+\langle
S_{\Phi}^{-1}\phi_{j},\phi_{j}\rangle S_{\Phi}^{-1}\phi_{j}$
$\displaystyle=\|S_{\Phi}^{-1/2}\phi_{i}\|^{2}S_{\Phi}^{-1}\phi_{i}+\langle
S_{\Phi}^{-1/2}\phi_{j},S_{\Phi}^{-1/2}\phi_{i}\rangle S_{\Phi}^{-1}\phi_{i}$
$\displaystyle\qquad\qquad+\langle
S_{\Phi}^{-1/2}\phi_{i},S_{\Phi}^{-1/2}\phi_{j}\rangle
S_{\Phi}^{-1}\phi_{j}+\|S_{\Phi}^{-1/2}\phi_{j}\|^{2}S_{\Phi}^{-1}\phi_{j}$
$\displaystyle=\|D_{1}^{-1/2}D_{1}M_{1}^{*}(e_{i})\|^{2}S_{\Phi}^{-1}\phi_{i}+\langle
D_{1}^{-1/2}D_{1}M_{1}^{*}(e_{j}),D_{1}^{-1/2}D_{1}M_{1}^{*}(e_{i})\rangle
S_{\Phi}^{-1}\phi_{i}$ $\displaystyle\qquad\qquad+\langle
D_{1}^{-1/2}D_{1}M_{1}^{*}(e_{i}),D_{1}^{-1/2}D_{1}M_{1}^{*}(e_{j})\rangle
S_{\Phi}^{-1}\phi_{j}$
$\displaystyle\qquad\qquad\qquad+\|D_{1}^{-1/2}D_{1}M_{1}^{*}(e_{j})\|^{2}S_{\Phi}^{-1}\phi_{j}$
$\displaystyle=\|M_{1}^{*}(e_{i})\|^{2}S_{\Phi}^{-1}\phi_{i}+\langle
M_{1}^{*}(e_{j}),M_{1}^{*}(e_{i})\rangle S_{\Phi}^{-1}\phi_{i}$
$\displaystyle\qquad\qquad+\langle M_{1}^{*}(e_{i}),M_{1}^{*}(e_{j})\rangle
S_{\Phi}^{-1}\phi_{j}+\|M_{1}^{*}(e_{j})\|^{2}S_{\Phi}^{-1}\phi_{j}$
$\displaystyle=\left(1-\frac{1}{n}\right)S_{\Phi}^{-1}\phi_{i}-\frac{1}{n}S_{\Phi}^{-1}\phi_{i}-\frac{1}{n}S_{\Phi}^{-1}\phi_{j}+\left(1-\frac{1}{n}\right)S_{\Phi}^{-1}\phi_{j}$
$\displaystyle=\frac{n-2}{n}\left(S_{\Phi}^{-1}\phi_{i}+S_{\Phi}^{-1}\phi_{j}\right).$

Thus, $\frac{n-2}{n}$ is an eigenvalue of
$E_{\Phi,S_{\Phi}^{-1}\Phi,\Lambda}$. Similarly,
$E_{\Phi,S_{\Phi}^{-1}\Phi,\Lambda}(S_{\Phi}^{-1}\phi_{i}-S_{\Phi}^{-1}\phi_{j})=\left(1-\frac{1}{n}+\frac{1}{n}\right)\break\left(S_{\Phi}^{-1}\phi_{i}-S_{\Phi}^{-1}\phi_{j}\right)=S_{\Phi}^{-1}\phi_{i}-S_{\Phi}^{-1}\phi_{j}$.
Then, the spectrum of $E_{\Phi,S_{\Phi}^{-1}\Phi,\Lambda}$ is
$\left\\{1,\frac{n-2}{n}\right\\}$. Hence,
$\rho(E_{\Phi,S_{\Phi}^{-1}\Phi,\Lambda})=1$ for any $\Lambda\subset[n]$ such
that $|\Lambda|=2$. Thus, $\rho_{\Phi,S_{\Phi}^{-1}\Phi}^{(2)}=1$. ∎

In the following theorem, we show that the canonical dual frame of a frame
generated by a connected graph is the unique spectrally optimal dual frame for
$2$-erasures.

######  Theorem 4.2.

Let $G$ be a connected graph on $n$ vertices and
$\Phi=\\{\phi_{i}\\}_{i\in[n]}$ be an $L_{G}(n,n-1)$-frame for
$\mathbb{C}^{n-1}$ with frame operator $S_{\Phi}$. If
$\Psi=\\{\psi_{i}\\}_{i\in[n]}$ is a dual frame of $\Phi$ then
$\rho_{\Phi,\Psi}^{(2)}\geq 1$. Further, $\rho_{\Phi}^{(2)}=1$ and
$S_{\Phi}^{-1}\Phi$ is the unique $SOD$-frame for $2$-erasures.

###### Proof.

Let $L$ be the Laplacian matrix of $G$. Since $\Phi$ is an
$L_{G}(n,n-1)$-frame, there exist an orthogonal matrix $M$ and diagonal matrix
$D=diag(\lambda_{1},\lambda_{2},\ldots,\lambda_{n-1},0)$ such that $L=MDM^{*}$
and $\Phi=D_{1}M_{1}^{*}$ where $M_{1}$ is formed by considering the first
$n-1$ columns of $M$ and
$D_{1}=diag(\sqrt{\lambda_{1}},\sqrt{\lambda_{2}},\ldots,\sqrt{\lambda_{n-1}})$.
Let $S_{\Phi}$ be the frame operator of $\Phi$ and
$\Psi=\\{\psi_{i}\\}_{i\in[n]}$ be any dual frame of $\Phi$. Then by Theorem
2.5, there exist $\mu\in\mathbb{C}^{n-1}$ such that
$\psi_{i}=S_{\Phi}^{-1}\phi_{i}+\mu$ for $i\in[n]$. Let
$\Lambda=\\{i,j\\}\subset[n]$ be arbitrary. Then

$\displaystyle E_{\Phi,\Psi,\Lambda}(\psi_{i}-\psi_{j})$
$\displaystyle=\langle S_{\Phi}^{-1}\phi_{i}+\mu-
S_{\Phi}^{-1}\phi_{j}-\mu,\phi_{i}\rangle(S_{\Phi}^{-1}\phi_{i}+\mu)$
$\displaystyle\qquad\qquad\qquad+\langle S_{\Phi}^{-1}\phi_{i}+\mu-
S_{\Phi}^{-1}\phi_{j}-\mu,\phi_{j}\rangle(S_{\Phi}^{-1}\phi_{j}+\mu)$
$\displaystyle=(\|S_{\Phi}^{-1/2}\phi_{i}\|^{2}-\langle
S_{\Phi}^{-1/2}\phi_{j},S_{\Phi}^{-1/2}\phi_{i}\rangle)(S_{\Phi}^{-1}\phi_{i}+\mu)$
$\displaystyle\qquad\qquad+(\langle
S_{\Phi}^{-1/2}\phi_{i},S_{\Phi}^{-1/2}\phi_{j}\rangle-\|S_{\Phi}^{-1/2}\phi_{j}\|^{2})(S_{\Phi}^{-1}\phi_{j}+\mu)$
$\displaystyle=(\|M_{1}^{*}(e_{i})\|^{2}-\langle
M_{1}^{*}(e_{j}),M_{1}^{*}(e_{i})\rangle)(S_{\Phi}^{-1}\phi_{i}+\mu)$
$\displaystyle\qquad\qquad+(\langle
M_{1}^{*}(e_{i}),M_{1}^{*}(e_{j})\rangle-\|M_{1}^{*}(e_{j})\|^{2})(S_{\Phi}^{-1}\phi_{j}+\mu)$
$\displaystyle=\left(1-\frac{1}{n}+\frac{1}{n}\right)(S_{\Phi}^{-1}\phi_{i}+\mu)+\left(-\frac{1}{n}-1+\frac{1}{n}\right)(S_{\Phi}^{-1}\phi_{j}+\mu)$
$\displaystyle=\psi_{i}-\psi_{j}.$

Thus, $1$ is an eigenvalue of $E_{\Phi,\Psi,\Lambda}$. Then
$\rho(E_{\Phi,\Psi,\Lambda})\geq 1$. Hence,
$\rho^{(2)}_{\Phi,\Psi}=\max\\{\rho(E_{\Phi,\Psi,\Lambda}):\Lambda\subset[n]\text{
and }|\Lambda|=2\\}\geq 1$. Thus, by Theorem 3.2 and Theorem 4.1,
$S_{\Phi}^{-1}\Phi$ is the unique $SOD$-frame for $2$-erasures. ∎

######  Corollary 4.1.

Let $G$ be a connected graph on $n$ vertices and
$\Phi=\\{\phi_{i}\\}_{i\in[n]}$ be an $G(n,n-1)$-frame for $\mathbb{C}^{n-1}$
with frame operator $S_{\Phi}$. Then canonical dual of $\Phi$ is the unique
$SOD$-frame for $2$-erasures and $\rho^{(2)}_{\Phi}=2$.

In the next theorem, we show that if $\Phi$ is an $L_{G}(n,k)$-frame generated
by any graph $G$ (need not be connected) with frame operator $S_{\Phi}$ then
$\rho_{\Phi,S_{\Phi}^{-1}\Phi}^{(2)}=1$.

######  Theorem 4.3.

Let $G$ be a simple graph on $n$ vertices with $k$ components and
$\Phi=\\{\phi_{i}\\}_{i\in[n]}$ is an $L_{G}(n,n-k)$-frame for
$\mathbb{C}^{k}$. If $S_{\Phi}$ is the frame operator of $\Phi$ then
$\rho_{\Phi,S_{\Phi}^{-1}\Phi}^{(2)}=1$.

###### Proof.

Let $G_{1},G_{2},\ldots,G_{k}$ be the components of $G$ and
$|V(G_{i})|=n_{i}$, for $i\in[k]$. Suppose
$V(G_{i})=\\{l_{i-1}+1,l_{i-1}+2,\ldots,l_{i}\\}$ where
$l_{j}=n_{1}+\cdots+n_{j}$ for $j\in[k]$ and $l_{0}=0$. For $i\in[k]$, let
$\lambda_{i,1}\geq\lambda_{i,2}\geq\cdots\geq\lambda_{i,n_{i}-1}>\lambda_{i,n_{i}}=0$
be the eigenvalues of $L(G_{i})$. Then there exist an orthogonal matricx
$M_{i}$ such that $L(G_{i})=M_{i}D_{i}M_{i}^{*}$ where
$D_{i}=diag(\lambda_{i,1},\lambda_{i,2},\ldots,\lambda_{i,n_{i}-1},0)$, for
$i\in[k]$, and $L(G)=MDM^{*}$ where $M=M_{1}\oplus\cdots\oplus M_{k}$ and
$D=D_{1}\oplus\cdots\oplus D_{k}$. Thus, the synthesis operator of an
$L_{G}(n,n-k)$-frame, say
$\widetilde{\Phi}=\\{\widetilde{\phi}_{i}\\}_{i\in[n]}$ is $B=B_{1}\oplus
B_{2}\oplus\cdots\oplus B_{k}$ such that
$B_{j}=diag(\sqrt{\lambda_{j,1}},\sqrt{\lambda_{j,2}},\ldots,\sqrt{\lambda_{j,n_{j}-1}})\widetilde{M_{j}}^{*}$
(where $\widetilde{M_{j}}^{*}$ is formed by taking the first $n_{j}-1$ columns
of $M_{j}$) is the synthesis operator of the $L_{G_{i}}(n_{i},n_{i}-1)$-frame,
say $\Phi_{j}=\\{\phi_{j,i}\\}_{i\in[n_{j}]}$ for $\mathbb{C}^{n_{j}-1}$, for
$j\in[k]$. By Theorem 2.3, there is a unitary operator $V$ such that
$\phi_{i}=V\widetilde{\phi}_{i}$ for all $i\in[n]$. Let $S_{\Phi}$ and
$S_{\widetilde{\Phi}}$ be the frame operators of $\Phi$ and
$\widetilde{\Phi}$, respectively. For $i\in[k]$, let $S_{\Phi_{i}}$ be the
frame operator of $\Phi_{i}$. Then

$\displaystyle S_{\widetilde{\Phi}}$ $\displaystyle=BB^{*}$
$\displaystyle=B_{1}B_{1}^{*}\oplus\cdots\oplus B_{k}B_{k}^{*}$
$\displaystyle=S_{\Phi_{1}}\oplus\cdots\oplus S_{\Phi_{k}}.$

Thus, $S_{\widetilde{\Phi}}^{-1}=S_{\Phi_{1}}^{-1}\oplus\cdots\oplus
S_{\Phi_{k}}^{-1}$, and $S_{\Phi}^{-1}=VS_{\widetilde{\Phi}}^{-1}V^{*}$. Let
$\Lambda=\\{p,q\\}\subset[n]$ be arbitrary. Now we proceed in the following
cases.

Case I:  Let $\Lambda\subset V(G_{j})$ for some $j\in[k]$. Suppose
$f=\left[\begin{array}[]{c}f_{1}\\\ f_{2}\\\ \vdots\\\
f_{k}\end{array}\right]\in\mathbb{C}^{n-k}$ where
$f_{i}\in\mathbb{C}^{n_{i}-1}$. We have

$\displaystyle
E_{\widetilde{\Phi},S_{\widetilde{\Phi}}^{-1}\widetilde{\Phi},\Lambda}(f)$
$\displaystyle=\langle f,\widetilde{\phi}_{p}\rangle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{p}+\langle
f,\widetilde{\phi}_{q}\rangle S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{q}$
$\displaystyle=\langle f_{j},\widetilde{\phi}_{j,p-l_{j-1}}\rangle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{p}+\langle
f_{j},\widetilde{\phi}_{j,q-l_{j-1}}\rangle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{q}$
$\displaystyle=\left[\begin{array}[]{c}0\\\ \vdots\\\ 0\\\
E_{\Phi_{j},S_{\Phi_{j}}^{-1}\Phi_{j},\Lambda^{\prime}}(f_{j})\\\ 0\\\
\vdots\\\ 0\end{array}\right],\text{ where
}\Lambda^{\prime}=\\{p-l_{j-1},q-l_{j-1}\\}.$

Thus, by Theorem 4.1,
$\rho(E_{\widetilde{\Phi},S_{\widetilde{\Phi}}^{-1}\widetilde{\Phi},\Lambda})=\rho(E_{\Phi_{j},S_{\Phi_{j}}^{-1}\Phi_{j},\Lambda^{\prime}})=1$.

Case II:  Let $p\in V(G_{s})$ and $q\in V(G_{r})$ where $s\neq r$. Then, we
have

$\displaystyle
E_{\widetilde{\Phi},S_{\widetilde{\Phi}}^{-1}\widetilde{\Phi},\Lambda}(S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{p})$
$\displaystyle=\langle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{p},\widetilde{\phi}_{p}\rangle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{p}+\langle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{p},\widetilde{\phi}_{q}\rangle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{q}$
$\displaystyle=\|S_{\widetilde{\Phi}}^{-1/2}\widetilde{\phi}_{p}\|^{2}S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{p}+0(S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{q})$
$\displaystyle=\frac{n_{s}-1}{n_{s}}S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{p}.$

Similarly,
$E_{\widetilde{\Phi},S_{\widetilde{\Phi}}^{-1}\widetilde{\Phi},\Lambda}(S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{q})=\frac{n_{r}-1}{n_{r}}S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{q}$.
Thus,
$\rho(E_{\widetilde{\Phi},S_{\widetilde{\Phi}}^{-1}\widetilde{\Phi},\Lambda})=\max\\{\frac{n_{s}-1}{n_{s}},\frac{n_{r}-1}{n_{r}}\\}<1$.

Hence, by Case I and Case II,
$\max\\{\rho(E_{\widetilde{\Phi},S_{\widetilde{\Phi}}^{-1}\widetilde{\Phi},\Lambda}):\Lambda\subset[n]\text{
and }|\Lambda|=2\\}=1$. Therefore, we have

$\displaystyle\rho^{(2)}_{\Phi,S_{\Phi}^{-1}\Phi}$
$\displaystyle=\max\\{\rho(E_{\Phi,S_{\Phi}^{-1}\Phi,\Lambda}):\Lambda\subset[n]\text{
and }|\Lambda|=2\\}$
$\displaystyle=\max\\{\rho(VE_{\widetilde{\Phi},S_{\widetilde{\Phi}}^{-1}\widetilde{\Phi},\Lambda}V^{*}):\Lambda\subset[n]\text{
and }|\Lambda|=2\\}$
$\displaystyle=\max\\{\rho(E_{\widetilde{\Phi},S_{\widetilde{\Phi}}^{-1}\widetilde{\Phi},\Lambda}):\Lambda\subset[n]\text{
and }|\Lambda|=2\\}=1.$

∎

In Theorem 3.3, we show that the canonical dual frames of a frame generated by
disconnected graphs are non-unique $SOD$-frames for $1$-erasure. Hence, it
would be interesting to see whether canonical dual frames of frames generated
by disconnected graphs are $SOD$-frames for $2$-erasures or not. In the next
two theorems, we show that canonical duals of $L_{G}(n,k)$-frames, where $G$
is a disconnected graph, are non-unique $SOD$-frames for $2$-erasures.

######  Theorem 4.4.

Suppose $G$ is a graph with $n$ vertices and $k>1$ components. If
$\Phi=\\{\phi_{i}\\}_{i\in[n]}$ is an $L_{G}(n,n-k)$-frame for
$\mathbb{C}^{n-k}$ with frame operator $S_{\Phi}$, then the canonical dual
frame $S_{\Phi}^{-1}\Phi$ of $\Phi$ is a spectrally optimal dual frame of
$\Phi$ for $2$-erasures. Further, $\rho^{(2)}_{\Phi}=1$.

###### Proof.

Let $G_{1},G_{2},\ldots,G_{k}$ be the connected components of $G$ such that
$|V(G_{i})|=n_{i}$ for $i\in[k]$. Let the vertex set of $G_{i}$ be
$\\{l_{i-1}+1,l_{i-1}+2,\ldots,l_{i}\\}$ for $i\in[k]$ where $l_{0}=0$ and for
$1\leq i\leq k$, $l_{i}=n_{1}+n_{2}+\cdots+n_{i}$. Let
$\Psi=\\{\psi_{i}\\}_{i\in[n]}$ be any dual frame of $\Phi$. Then by Theorem
2.5, there exist $\mu_{1},\mu_{2},\ldots,\mu_{k}\in\mathbb{C}^{n-k}$ such that
$\Psi=\bigcup\limits_{j\in[k]}\\{S_{\Phi}^{-1}\phi_{l_{j-1}+i}+\mu_{j}\\}_{i\in[n_{j}]}$.
Without loss of generality, assume that $|V(G_{1})|\geq 2$. Take
$\Lambda=\\{1,2\\}$. Then, $E_{\Phi,\Psi,\Lambda}(f)=\langle
f,\phi_{1}\rangle(S_{\Phi}^{-1}\phi_{1}+\mu_{1})+\langle
f,\phi_{2}\rangle(S_{\Phi}^{-1}\phi_{2}+\mu_{1})$. Thus, we have

$\displaystyle
E_{\Phi,\Psi,\Lambda}(S_{\Phi}^{-1}\phi_{1}+\mu_{1}-(S_{\Phi}^{-1}\phi_{2}+\mu_{1}))$
$\displaystyle=\langle
S_{\Phi}^{-1}\phi_{1}+\mu_{1}-S_{\Phi}^{-1}\phi_{2}-\mu_{1},\phi_{1}\rangle(S_{\Phi}^{-1}\phi_{1}+\mu_{1})$
$\displaystyle\qquad\qquad+\langle
S_{\Phi}^{-1}\phi_{1}+\mu_{1}-S_{\Phi}^{-1}\phi_{2}-\mu_{1},\phi_{2}\rangle(S_{\Phi}^{-1}\phi_{2}+\mu_{1})$
$\displaystyle=(\langle S_{\Phi}^{-1}\phi_{1},\phi_{1}\rangle-\langle
S_{\Phi}^{-1}\phi_{2},\phi_{1}\rangle)(S_{\Phi}^{-1}\phi_{1}+\mu_{1})$
$\displaystyle\qquad\qquad+(\langle
S_{\Phi}^{-1}\phi_{1},\phi_{2}\rangle-\langle
S_{\Phi}^{-1}\phi_{2},\phi_{2}\rangle)(S_{\Phi}^{-1}\phi_{2}+\mu_{1})$
$\displaystyle=(\|S_{\Phi}^{-1/2}\phi_{1}\|^{2}-\langle
S_{\Phi}^{-1/2}\phi_{2},S_{\Phi}^{-1/2}\phi_{1}\rangle)(S_{\Phi}^{-1}\phi_{1}+\mu_{1})$
$\displaystyle\qquad\qquad+(\langle
S_{\Phi}^{-1/2}\phi_{1},S_{\Phi}^{-1/2}\phi_{2}\rangle-\|S_{\Phi}^{-1/2}\phi_{2}\|^{2})(S_{\Phi}^{-1}\phi_{2}+\mu_{1})$
$\displaystyle=\left(1-\frac{1}{n_{1}}+\frac{1}{n_{1}}\right)(S_{\Phi}^{-1}\phi_{1}+\mu_{1})-\left(1-\frac{1}{n_{1}}+\frac{1}{n_{1}}\right)(S_{\Phi}^{-1}\phi_{2}+\mu_{1})$
$\displaystyle=(S_{\Phi}^{-1}\phi_{1}+\mu_{1})-(S_{\Phi}^{-1}\phi_{2}+\mu_{1}).$

Thus, $1$ is an eigenvalue of $E_{\Phi,\Psi,\Lambda}$. Hence,
$\rho(E_{\Phi,\Psi,\Lambda})\geq 1$. This gives that
$\rho^{(2)}_{\Phi,\Psi}\geq\rho(E_{\Phi,\Psi,\Lambda})\geq 1$. By Theorem 3.3,
$S_{\Phi}^{-1}\Phi$ is spectrally optimal dual frame for $1$-erasures and by
Theorem 4.3, $\rho^{(2)}_{\Phi,S_{\Phi}^{-1}\Phi}=1\geq\rho^{(2)}_{\Phi,\Psi}$
for any dual frame $\Psi$ of $\Phi$. Thus, the canonical dual frame of $\Phi$
is a spectrally optimal dual frame for $2$-erasures. ∎

In the next theorem, we show the existence of an alternate dual frame of a
frame generated by a disconnected graph such that the alternate dual is a
spectrally optimal dual frame for $2$-erasures.

######  Theorem 4.5.

Suppose $G$ is a graph with $n$ vertices and $k>1$ connected components. If
$\Phi=\\{\phi_{i}\\}_{i\in[n]}$ is an $L_{G}(n,n-k)$-frame for
$\mathbb{C}^{n-k}$ with frame operator $S_{\Phi}$, then there exist an
alternate dual frame of $\Phi$ such that it is a spectrally optimal dual frame
of $\Phi$ for $2$-erasures.

###### Proof.

Let $G_{1},G_{2},\ldots,G_{k}$ be the connected components of $G$ such that
$|V(G_{i})|=n_{i}$ for $i\in[k]$. Let the vertex set of $G_{i}$ be
$\\{v_{l_{i-1}+1},v_{l_{i-1}+2},\ldots,v_{l_{i}}\\}$ for $i\in[k]$ where
$l_{0}=0$ and for $i\geq 1$, $l_{i}=n_{1}+n_{2}+\cdots+n_{i}$. Let
$\widetilde{\Phi}$ be the $L_{G}(n,k)$-frame as obtained in Theorem 4.3. By
Theorem 4.4, $S_{\widetilde{\Phi}}^{-1}\widetilde{\Phi}$ is a spectrally
optimal dual frame of $\widetilde{\Phi}$ for $2$-erasures and
$\rho^{(2)}_{\widetilde{\Phi},S_{\widetilde{\Phi}}^{-1}\widetilde{\Phi}}=1$.
Let $\mu=[\mu_{1}\,,\mu_{2}\,,\ldots,\,\mu_{n-k}]^{t}\in\mathbb{C}^{n-k}$ such
that $\mu_{i}=\begin{cases}1,\,i=l_{1}+1\\\ 0,\,\text{otherwise}\end{cases}$.
Then
$\Psi=\\{\psi_{i}\\}_{i\in[n]}=\\{S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{i}+\mu\\}_{i\in[l_{1}]}\bigcup\\{S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{i}\\}_{i\in[n]\setminus[l_{1}]}$
is an alternate dual frame of $\widetilde{\Phi}$. Then

$\displaystyle\rho^{(1)}_{\widetilde{\Phi},\Psi}$
$\displaystyle=\max\\{|\langle\psi_{i},\widetilde{\phi}_{i}\rangle|:i\in[n]\\}$
$\displaystyle=\max\left(\\{|\langle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{i}+\mu,\widetilde{\phi}_{i}\rangle|:i\in[l_{1}]\\}\cup\\{|\langle
S_{\Phi}^{-1}\phi_{i},\widetilde{\phi}_{i}\rangle|:i\in[n]\setminus[l_{1}]\\}\right)$
$\displaystyle=\max\left(\\{|\langle
S_{\Phi}^{-1}\phi_{i},\widetilde{\phi}_{i}\rangle+\langle\mu,\widetilde{\phi}_{i}\rangle|:i\in[l_{1}]\\}\cup\\{|\langle
S_{\Phi}^{-1}\phi_{i},\widetilde{\phi}_{i}\rangle|:i\in[n]\setminus[l_{1}]\\}\right)$
$\displaystyle=\max\left(\\{|\langle
S_{\Phi}^{-1}\phi_{i},\widetilde{\phi}_{i}\rangle|:i\in[l_{1}]\\}\cup\\{|\langle
S_{\Phi}^{-1}\phi_{i},\widetilde{\phi}_{i}\rangle|:i\in[n]\setminus[l_{1}]\\}\right)=\rho^{(1)}_{\widetilde{\Phi},S^{-1}_{\widetilde{\Phi}}\widetilde{\Phi}}.$

Thus, by Theorem 3.3, $\Psi$ is a spectrally optimal dual frame of
$\widetilde{\Phi}$ for $1$-erasure. Let $\Lambda=\\{p,q\\}\subset[n]$ be
arbitrary. Then $E_{\widetilde{\Phi},\Psi,\Lambda}(f)=\langle
f,\widetilde{\phi}_{p}\rangle\psi_{p}+\langle
f,\widetilde{\phi}_{q}\rangle\psi_{q}$. Now we will proceed in cases.

Case I:  Let $\Lambda\subseteq V(G_{j})$ for some $j\in[k]$. Then, we have

$\displaystyle E_{\widetilde{\Phi},\Psi,\Lambda}(\psi_{p}+\psi_{q})$
$\displaystyle=\langle\psi_{p}+\psi_{q},\widetilde{\phi}_{p}\rangle\psi_{p}+\langle\psi_{p}+\psi_{q},\widetilde{\phi}_{q}\rangle\psi_{q}$
$\displaystyle=\begin{cases}\langle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{p}+S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{q}+2\mu,\widetilde{\phi}_{p}\rangle\psi_{p}+\langle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{p}+S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{p}+2\mu,\widetilde{\phi}_{q}\rangle\psi_{q}\text{
if }i=1\\\ \langle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{p}+S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{q},\widetilde{\phi}_{p}\rangle\psi_{p}+\langle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{p}+S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{q},\widetilde{\phi}_{q}\rangle\psi_{q}\text{
otherwise }\end{cases}$
$\displaystyle=\begin{cases}(\|S_{\widetilde{\Phi}}^{-1/2}\widetilde{\phi}_{p}\|^{2}+\langle
S_{\widetilde{\Phi}}^{-1/2}\widetilde{\phi}_{q},S_{\widetilde{\Phi}}^{-1/2}\widetilde{\phi}_{p}\rangle+2\langle\mu,\widetilde{\phi}_{p}\rangle)\psi_{p}\\\
\qquad\qquad+(\|S_{\widetilde{\Phi}}^{-1/2}\widetilde{\phi}_{q}\|^{2}+\langle
S_{\widetilde{\Phi}}^{-1/2}\widetilde{\phi}_{p},S_{\widetilde{\Phi}}^{-1/2}\widetilde{\phi}_{q}\rangle+2\langle\mu,\widetilde{\phi}_{q}\rangle)\psi_{q}\text{
if }i=1\\\ (\|S_{\widetilde{\Phi}}^{-1/2}\widetilde{\phi}_{p}\|^{2}+\langle
S_{\widetilde{\Phi}}^{-1/2}\widetilde{\phi}_{q},S_{\widetilde{\Phi}}^{-1/2}\widetilde{\phi}_{p}\rangle)\psi_{p}\\\
\qquad\qquad+(\|S_{\widetilde{\Phi}}^{-1/2}\widetilde{\phi}_{q}\|^{2}+\langle
S_{\widetilde{\Phi}}^{-1/2}\widetilde{\phi}_{p},S_{\widetilde{\Phi}}^{-1/2}\widetilde{\phi}_{q}\rangle)\psi_{q}\text{
otherwise }\end{cases}$
$\displaystyle=\begin{cases}\left(\left(1-\frac{1}{n_{1}}\right)-\frac{1}{n_{1}}+0\right)\psi_{p}+\left(\left(1-\frac{1}{n_{1}}\right)-\frac{1}{n_{1}}+0\right)\psi_{q}\text{
if }j=1\\\
\left(\left(1-\frac{1}{n_{j}}\right)-\frac{1}{n_{j}}\right)\psi_{p}+\left(\left(1-\frac{1}{n_{j}}\right)-\frac{1}{n_{j}}\right)\psi_{q}\text{
otherwise }\end{cases}$
$\displaystyle=\left(1-\frac{2}{n_{j}}\right)(\psi_{p}+\psi_{q}).$

Similarly,
$E_{\widetilde{\Phi},\Psi,\Lambda}(\psi_{p}-\psi_{q})=\psi_{p}-\psi_{q}$.
Thus, $\rho(E_{\widetilde{\Phi},\Psi,\Lambda})=1$.

Case II:  Let $p\in V(G_{r})$ and $q\in V(G_{s})$ where
$r,s\in[k]\setminus\\{1\\}$ and $r\neq s$. Then, we have

$\displaystyle E_{\widetilde{\Phi},\Psi,\Lambda}(\psi_{p})$
$\displaystyle=\langle\psi_{p},\widetilde{\phi}_{p}\rangle\psi_{p}+\langle\psi_{p},\widetilde{\phi}_{q}\rangle\psi_{q}$
$\displaystyle=\langle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{p},\widetilde{\phi}_{p}\rangle\psi_{p}+\langle
S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{p},\widetilde{\phi}_{q}\rangle\psi_{q}$
$\displaystyle=\|S_{\widetilde{\Phi}}^{-1/2}\widetilde{\phi}_{p}\|^{2}\psi_{p}+\langle
S_{\widetilde{\Phi}}^{-1/2}\widetilde{\phi}_{p},S_{\widetilde{\Phi}}^{-1/2}\widetilde{\phi}_{q}\rangle\psi_{q}$
$\displaystyle=\left(1-\frac{1}{n_{r}}\right)\psi_{p}+0$
$\displaystyle=\left(1-\frac{1}{n_{r}}\right)\psi_{p}.$

Similarly,
$E_{\widetilde{\Phi},\Psi,\Lambda}(\psi_{q})=\left(1-\frac{1}{n_{s}}\right)\psi_{q}$.
Hence,
$\rho(E_{\widetilde{\Phi},\Psi,\Lambda})=\max\left\\{1-\frac{1}{n_{r}},1-\frac{1}{n_{s}}\right\\}<1$.

Case III:  Let $p\in V(G_{1})$ and $q\in V(G_{r})$ where $r\neq 1$. Then
$E_{\widetilde{\Phi},\Psi,\Lambda}(f)=\langle
f,\widetilde{\phi}_{p}\rangle\psi_{p}+\langle
f,\widetilde{\phi}_{q}\rangle\psi_{q}=\langle
f,\widetilde{\phi}_{p}\rangle(S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{p}+\mu)+\langle
f,\widetilde{\phi}_{q}\rangle S_{\widetilde{\Phi}}^{-1}\widetilde{\phi}_{q}$.
The matrix representation of
$E_{\widetilde{\Phi},S_{\widetilde{\Phi}}^{-1}\widetilde{\Phi},\Lambda}$ with
respect to the standard canonical orthonormal basis is of the form,
$[E_{\widetilde{\Phi},S_{\widetilde{\Phi}}^{-1}\widetilde{\Phi},\Lambda}]=\left[\begin{array}[]{cc}E_{1}&0\\\
0&E_{2}\end{array}\right]$ for some matrices $E_{1}$ of order $n_{1}-1$ and
$E_{2}$ of order $(n-k)-(n_{1}-1)$. Then, the matrix representation of
$E_{\widetilde{\Phi},\Psi,\Lambda}$ with respect to the standard canonical
orthonormal basis is
$[E_{\widetilde{\Phi},\Psi,\Lambda}]=\left[\begin{array}[]{cc}E_{1}&0\\\
E_{3}&E_{2}\end{array}\right]$ where
$E_{3}=\left[\begin{array}[]{cccc}\alpha_{1}&\alpha_{2}&\cdots&\alpha_{l_{1}-1}\\\
0&0&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&0\end{array}\right]$
and $\alpha_{i}$ is the $i^{th}$-component of $\widetilde{\phi}_{p}$. Thus,
$\rho(E_{\widetilde{\Phi},\Psi,\Lambda})=\rho([E_{\widetilde{\Phi},S_{\widetilde{\Phi}}^{-1}\widetilde{\Phi},\Lambda}])=1$.
Then, by Theorem 4.4,
$\rho^{(2)}_{\widetilde{\Phi},\Psi}=1=\rho^{(2)}_{\widetilde{\Phi}}$. Hence,
$\Psi$ is an alternate dual frame of $\widetilde{\Phi}$ such that $\Psi$ is a
spectrally optimal dual frame of $\widetilde{\Phi}$ for $2$-erasures.
Therefore, by Theorem 3.1, there exists an alternate dual frame of $\Phi$,
which is an $SOD$-frame of $\Phi$ for $2$-erasures. ∎

######  Corollary 4.2.

Suppose $G$ is a graph with $n$ vertices and $k>1$ connected components. If
$\Phi=\\{\phi_{i}\\}_{i\in[n]}$ is an $G(n,n-k)$-frame for $\mathbb{C}^{n-k}$
with frame operator $S_{\Phi}$, then there exist an alternate dual frame of
$\Phi$ such that it is an $SOD$-frame of $\Phi$ for $2$-erasures.

In the following example, we show explicitly the existence of an alternate
dual frame that is an $SOD$-frame for $2$-erasures of a frame generated by a
disconnected graph.

######  Example 4.1.

Consider the frame $Phi=\\{\phi_{i}\\}_{i\in[5]}$ generated by graph $G$ given
in Example 3.1. The canonical dual frame $S_{\Phi}^{-1}\Phi$ is an $SOD$-frame
of $\Phi$ for $1$-erasure. Now, we will show that $S_{\Phi}^{-1}\Phi$ is a
non-unique $SOD$-frame of $\Phi$ for $2$-erasures. First, we give the matrix
representation of error operators. For $\Lambda=\\{i,j\\}\subset[5]$,
$E_{\Phi,S_{\Phi}^{-1}\Phi,\Lambda}(f)=\langle f,\phi_{i}\rangle
S_{\Phi}^{-1}\phi_{i}+\langle f,\phi_{j}\rangle S_{\Phi}^{-1}\phi_{j}$. We
have

$\left[E_{\Phi,S_{\Phi}^{-1}\Phi,\\{1,2\\}}\right]=\left[\begin{array}[]{ccc}\frac{1}{2}&\frac{-1}{2\sqrt{3}}&0\\\\[2.84526pt]
\frac{-1}{2\sqrt{3}}&\frac{5}{6}&0\\\\[2.84526pt]
0&0&0\end{array}\right],\left[E_{\Phi,S_{\Phi}^{-1}\Phi,\\{1,3\\}}\right]=\left[\begin{array}[]{ccc}1&0&0\\\\[2.84526pt]
0&\frac{1}{3}&0\\\\[2.84526pt]
0&0&0\end{array}\right],\left[E_{\Phi,S_{\Phi}^{-1}\Phi,\\{2,3\\}}\right]=\left[\begin{array}[]{ccc}\frac{1}{2}&\frac{1}{2\sqrt{3}}&0\\\\[2.84526pt]
\frac{1}{2\sqrt{3}}&\frac{5}{6}&0\\\\[2.84526pt] 0&0&0\end{array}\right],$
$\left[E_{\Phi,S_{\Phi}^{-1}\Phi,\\{4,5\\}}\right]=\left[\begin{array}[]{ccc}0&0&0\\\\[2.84526pt]
0&0&0\\\\[2.84526pt]
0&0&1\end{array}\right],\left[E_{\Phi,S_{\Phi}^{-1}\Phi,\\{1,4\\}}\right]=\left[E_{\Phi,S_{\Phi}^{-1}\Phi,\\{1,5\\}}\right]=\left[\begin{array}[]{ccc}\frac{1}{2}&\frac{-1}{2\sqrt{3}}&0\\\\[2.84526pt]
\frac{-1}{2\sqrt{3}}&\frac{1}{6}&0\\\\[2.84526pt]
0&0&\frac{1}{2}\end{array}\right],$
$\left[E_{\Phi,S_{\Phi}^{-1}\Phi,\\{2,4\\}}\right]=\left[E_{\Phi,S_{\Phi}^{-1}\Phi,\\{2,5\\}}\right]=\left[\begin{array}[]{ccc}0&0&0\\\\[2.84526pt]
0&\frac{2}{3}&0\\\\[2.84526pt]
0&0&\frac{1}{2}\end{array}\right],\left[E_{\Phi,S_{\Phi}^{-1}\Phi,\\{3,4\\}}\right]=\left[E_{\Phi,S_{\Phi}^{-1}\Phi,\\{3,5\\}}\right]=\left[\begin{array}[]{ccc}\frac{1}{2}&\frac{1}{2\sqrt{3}}&0\\\\[2.84526pt]
\frac{1}{2\sqrt{3}}&\frac{1}{6}&0\\\\[2.84526pt]
0&0&\frac{1}{2}\end{array}\right].$

Now
$\rho(E_{\Phi,S_{\Phi}^{-1}\Phi,\\{1,2\\}})=\rho(E_{\Phi,S_{\Phi}^{-1}\Phi,\\{1,3\\}})=\rho(E_{\Phi,S_{\Phi}^{-1}\Phi,\\{2,3\\}})=\max\left\\{1,\frac{1}{3},0\right\\}=1$,
$\rho(E_{\Phi,S_{\Phi}^{-1}\Phi,\\{4,5\\}})=\max\left\\{1,0\right\\}=1$ and
$\rho(E_{\Phi,S_{\Phi}^{-1}\Phi,\\{1,4\\}})=\rho(E_{\Phi,S_{\Phi}^{-1}\Phi,\\{1,5\\}})=\rho(E_{\Phi,S_{\Phi}^{-1}\Phi,\\{2,4\\}})=\rho(E_{\Phi,S_{\Phi}^{-1}\Phi,\\{2,5\\}})=\rho(E_{\Phi,S_{\Phi}^{-1}\Phi,\\{3,4\\}})=\rho(E_{\Phi,S_{\Phi}^{-1}\Phi,\\{3,5\\}})=\max\left\\{\frac{2}{3},\frac{1}{3},0\right\\}=\frac{2}{3}$.
Thus,
$\rho_{\Phi,S_{\Phi}^{-1}\Phi}^{(2)}=\max\left\\{1,\frac{2}{3}\right\\}=1$.
Let $\Psi=\\{\psi_{i}\\}_{i\in[5]}$ be any dual frame of $\Phi$. Then there
exist scalars $a_{o},b_{o},c_{o},e_{o},f_{o},g_{o}$ such that
$\Psi=\\{\psi_{i}\\}_{i\in[5]}=\left\\{\left[\begin{array}[]{c}\frac{1}{\sqrt{6}}+a_{o}\\\\[2.84526pt]
\frac{-1}{3\sqrt{2}}+b_{o}\\\\[2.84526pt]
c_{o}\end{array}\right],\left[\begin{array}[]{c}a_{o}\\\\[2.84526pt]
\frac{2}{3\sqrt{2}}+b_{o}\\\\[2.84526pt]
c_{o}\end{array}\right],\left[\begin{array}[]{c}\frac{-1}{\sqrt{6}}+a_{o}\\\\[2.84526pt]
\frac{-1}{3\sqrt{2}}+b_{o}\\\\[2.84526pt]
c_{o}\end{array}\right],\left[\begin{array}[]{c}e_{o}\\\\[2.84526pt]
f_{o}\\\\[2.84526pt]
\frac{1}{2}+g_{o}\end{array}\right],\left[\begin{array}[]{c}e_{o}\\\\[2.84526pt]
f_{o}\\\\[2.84526pt] \frac{-1}{2}+g_{o}\end{array}\right]\right\\}$.
For $\Lambda=\\{1,2\\}$, $E_{\Phi,\Psi,\Lambda}(f)=\langle
f,\phi_{1}\rangle\psi_{1}+\langle f,\phi_{2}\rangle\psi_{2}$. For
$f=\left[\begin{array}[]{c}\frac{1}{\sqrt{6}}\\\\[2.84526pt]
\frac{-1}{\sqrt{2}}\\\\[2.84526pt] 0\end{array}\right]$,
$E_{\Phi,\Psi,\Lambda}(f)=\left[\begin{array}[]{c}\frac{1}{\sqrt{6}}+a_{o}\\\\[2.84526pt]
\frac{-1}{3\sqrt{2}}+b_{o}\\\\[2.84526pt]
c_{o}\end{array}\right]-\left[\begin{array}[]{c}a_{o}\\\\[2.84526pt]
\frac{2}{3\sqrt{2}}+b_{o}\\\\[2.84526pt]
c_{o}\end{array}\right]=\left[\begin{array}[]{c}\frac{1}{\sqrt{6}}\\\\[2.84526pt]
\frac{-1}{\sqrt{2}}\\\\[2.84526pt] 0\end{array}\right]$. Thus, $1$ is an
eigenvalue of $E_{\Phi,\Psi,\Lambda}$ and hence
$\rho(E_{\Phi,\Psi,\Lambda})\geq 1$. Therefore, $\rho^{(2)}_{\Phi,\Psi}\geq
1=\rho^{(2)}_{\Phi,S^{-1}_{\Phi}\Phi}$ for any dual frame $\Psi$ of the frame
$\Phi$. Thus, $S^{-1}_{\Phi}\Phi$ is an $SOD$-frame of $\Phi$ for
$2$-erasures. Consider the dual frame $\Psi_{1}$ given in Example 3.1. It is
shown in Example 3.1 that $\Psi_{1}$ is an $SOD$-frame of $\Phi$ for
$1$-erasure. Further, we have

$\left[E_{\Phi,\Psi_{1},\\{1,2\\}}\right]=\left[\begin{array}[]{ccc}\frac{1}{2}&\frac{-1}{2\sqrt{3}}&0\\\
\frac{-1}{2\sqrt{3}}&\frac{5}{6}&0\\\
\frac{\sqrt{3}}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\end{array}\right],\left[E_{\Phi,\Psi_{1},\\{1,3\\}}\right]=\left[\begin{array}[]{ccc}1&0&0\\\
0&\frac{1}{3}&0\\\
0&-\sqrt{2}&0\end{array}\right],\left[E_{\Phi,\Psi_{1},\\{2,3\\}}\right]=\left[\begin{array}[]{ccc}\frac{1}{2}&\frac{1}{2\sqrt{3}}&0\\\
\frac{1}{2\sqrt{3}}&\frac{5}{6}&0\\\
\frac{-\sqrt{3}}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\end{array}\right],$
$\left[E_{\Phi,\Psi_{1},\\{4,5\\}}\right]=\left[\begin{array}[]{ccc}0&0&0\\\
0&0&0\\\
0&0&1\end{array}\right],\left[E_{\Phi,\Psi_{1},\\{1,4\\}}\right]=\left[E_{\Phi,\Psi_{1},\\{1,5\\}}\right]=\left[\begin{array}[]{ccc}\frac{1}{2}&\frac{-1}{2\sqrt{3}}&0\\\
\frac{-1}{2\sqrt{3}}&\frac{1}{6}&0\\\
\frac{\sqrt{3}}{\sqrt{2}}&\frac{1}{\sqrt{2}}&\frac{1}{2}\end{array}\right],$
$\left[E_{\Phi,\Psi_{1},\\{2,4\\}}\right]=\left[E_{\Phi,\Psi_{1},\\{2,5\\}}\right]=\left[\begin{array}[]{ccc}0&0&0\\\
0&\frac{2}{3}&0\\\
0&\sqrt{2}&\frac{1}{2}\end{array}\right],\left[E_{\Phi,\Psi_{1},\\{3,4\\}}\right]=\left[E_{\Phi,\Psi_{1},\\{3,5\\}}\right]=\left[\begin{array}[]{ccc}\frac{1}{2}&\frac{1}{2\sqrt{3}}&0\\\
\frac{1}{2\sqrt{3}}&\frac{1}{6}&0\\\
\frac{-\sqrt{3}}{\sqrt{2}}&\frac{-1}{\sqrt{2}}&\frac{1}{2}\end{array}\right].$

Now
$\rho(E_{\Phi,\Psi_{1},\\{1,2\\}})=\rho(E_{\Phi,\Psi_{1},\\{1,3\\}})=\rho(E_{\Phi,\Psi_{1},\\{2,3\\}})=\max\left\\{1,\frac{1}{3},0\right\\}=1$,
$\rho(E_{\Phi,\Psi_{1},\\{4,5\\}})=\max\left\\{1,0\right\\}=1$ and
$\rho(E_{\Phi,\Psi_{1},\\{1,4\\}})=\rho(E_{\Phi,\Psi_{1},\\{1,5\\}})=\rho(E_{\Phi,\Psi_{1},\\{2,4\\}})=\rho(E_{\Phi,\Psi_{1},\\{2,5\\}})=\rho(E_{\Phi,\Psi_{1},\\{3,4\\}})=\rho(E_{\Phi,\Psi_{1},\\{3,5\\}})=\max\left\\{\frac{2}{3},\frac{1}{3},0\right\\}=\frac{2}{3}$.
Thus, $\rho_{\Phi,\Psi_{1}}^{(2)}=\max\left\\{1,\frac{2}{3}\right\\}=1$.
Hence, $\Psi_{1}$ is an $SOD$-frame of $\Phi$ for $2$-erasures. Therefore, the
canonical dual is a non-unique $SOD$-frame of $\Phi$ for $2$-erasures.

## Acknowledgments

Aniruddha Samanta expresses thanks to the National Board for Higher
Mathematics (NBHM), Department of Atomic Energy, India, for providing
financial support in the form of an NBHM Post-doctoral Fellowship (Sanction
Order No. 0204/21/2023/R&D-II/10038). The second author also acknowledges
excellent working conditions in the Theoretical Statistics and Mathematics
Unit, Indian Statistical Institute Kolkata.

## References

  * [1] F. Arabyani-Neyshaburi, A. A. Arefijamaal and G. Sadeghi, Numerically and spectrally optimal dual frames in Hilbert spaces, _Linear Algebra Appl._ , 604(2020), 52–-71.
  * [2] R. B. Bapat, Graphs and matrices, Springer, London; Hindustan Book Agency, New Delhi, 2010.
  * [3] R. B. Bapat, D. Kalita and S. Pati, On weighted directed graphs, _Linear Algebra Appl._ , 436(1)(2012),99–111.
  * [4] J. Benedetto and S. Li, The theory of multiresolution analysis frames and applications to filter banks, _Appl. Comp. Harm. Anal._ , 5(1998), 389–427.
  * [5] N. Biggs, Algebraic graph theory, Cambridge Mathematical Library, second edn, Cambridge University Press, Cambridge, 1993.
  * [6] B. G. Bodmann and V. I. Paulsen, Frames, graphs and erasures, _Linear Algebra Appl._ , 404(2005), 118–146.
  * [7] P. G. Casazza and J. Kovačević, Equal-norm tight frames with erasures, _Adv. Comput. Math._ , 18(2003), 387–430.
  * [8] P. G. Casazza and G. Kutyniok, Finite Frames: Theory and Applications, Birkh$\ddot{a}$user, 2012.
  * [9] O. Christensen, An introduction to frames and Riesz bases, Second edition, Birkh$\ddot{a}$user, 2016.
  * [10] D. Cvetkovi´c and S. Simi´c, Graph spectra in computer science, _Linear Algebra Appl._ , 434(6)(2011), 1545–1562.
  * [11] I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, _J. Math Phys._ , 27(1986), 1271–1283.
  * [12] Deepshikha, Frames generated by graphs, arXiv preprint, arxiv: 2405.16891, 2024.
  * [13] Deepshikha and A. Samanta, Averaged numerically optimal dual frames for erasures, _Linear Multilinear Algebra_ , 71(2)(2023), 301–316.
  * [14] Deepshikha and A. Samanta, On walk-regular graphs and optimal duals of frames generated by graphs, arXiv preprint, arxiv: 2405.18189, 2024.
  * [15] R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, _Trans. Amer. Math. Soc._ , 72(1952), 341–366.
  * [16] K. Guo and B. Mohar, Hermitian adjacency matrix of digraphs and mixed graphs, _J. Graph Theory_ , 85(1)(2017),217–248.
  * [17] R. Holmes and V. Paulsen, Optimal frames for erasures, _Linear Algebra Appl._ , 377(2004), 31-–51.
  * [18] J. Lopez and D. Han, Optimal dual frames for erasures, _Linear Algebra Appl._ , 432(1)(2010), 471–-482.
  * [19] J. Leng and D. Han, Optimal dual frames for erasures II, _Linear Algebra Appl._ , 435(6)(2011), 1464–-1472.
  * [20] R. Mehatari, M. R. Kannan and A. Samanta, On the adjacency matrix of a complex unit gain graph, _Linear Multilinear Algebra_ , 70(9)(2020), 1798–1813.
  * [21] N. Nisan and A. Wigderson, _On rank vs. communication complexity_ , Combinatorica 15(4)(1995), 557–565.
  * [22] S. Pehlivan, D. Han and R. Mohapatra, Linearly connected sequences and spectrally optimal dual frames for erasures, _J. Funct. Anal._ , 265(11)(2013), 2855–-2876.
  * [23] J. Shi and J. Malik, Normalized cuts and image segmentation, _IEEE Transactions on pattern analysis and machine intelligence_ , 22(8) (2000), 888–905.
  * [24] A. Samanta and M. R. Kannan, Gain distance matrices for complex unit gain graphs, _Discrete Math._ , 345(1)(2022), 112634.
  * [25] T. Zaslavsky, Signed graphs, _Discrete Appl. Math._ , 4(1)(1982),47–74.