# Interlacing Properties of Eigenvalues of Laplacian and Net-Laplacian Matrix
of Signed Graphs

Satyam Guragain Email<EMAIL_ADDRESS>Ravi Srivastava Corresponding
author, Email<EMAIL_ADDRESS>
###### Abstract

This paper explores interlacing inequalities in the Laplacian spectrum of
signed cycles and investigates interlacing relationship between the spectrum
of the net-Laplacian of a signed graph and its subgraph formed by removing a
vertex together with its incident edges. Additionally, an inequality is
derived between the net-Laplacian spectrum of a complete co-regular signed
graph $\Gamma$ and the Laplacian spectrum of the graph obtained by removing
any vertex $v$ from $\Gamma$. Also for a signed graph $\Gamma$, the net-
Laplacian matrix is normalized and an inequality is derived between the
spectrum of the normalized net-Laplacian of a signed graph and its subgraph,
formed by contraction of edge and vertex.

MSC 2020 Classifications: 05C22, 05C50, 15A42
Keywords: Signed graph, Laplacian matrix, net-Laplacian matrix, normalized
net-Laplacian, interlacing inequalities.

##  1 Introduction

All graphs considered in this paper are undirected, simple and finite. A graph
$G$ is defined as an ordered pair $G=(V,E)$ where $V$ represents the set of
vertices and $E$ represents the set of edges. In 1953 Frank Harary made
significant contributions by introducing the idea of signed graphs and the
concept of balancedness within such graphs [6]. A signed graph
$\Gamma=(G,\sigma)$ consists of a simple graph $G=(V,E)$ and a mapping
$\sigma:E\rightarrow\\{+,-\\}$ known as the signature of $\Gamma$ which assign
either positive or negative sign to edges.

The signed degree of a vertex $u$ (denoted by $sdeg(u)$) is the subtraction of
the negative degree $(d_{v}^{-})$ from the positive degree $(d_{v}^{+})$ while
the degree of the same vertex is simply the sum of its positive and negative
degrees. For a signed graph $\Gamma=(G,\sigma)$ with vertex set
$V(\Gamma)=\\{u_{1},u_{2},\cdots,u_{m}\\}$ the adjacency matrix of $\Gamma$ is
the $m\times m$ matrix given by $A(\Gamma)=(a_{lk}^{\sigma})$ where
$a_{lk}^{\sigma}=\sigma(v_{l}v_{k})a_{lk}$ and $a_{lk}=1$ if $v_{l}$ and
$v_{k}$ are adjacent and $a_{lk}=0$ otherwise. The Laplacian matrix and the
net-Laplacian matrix of $\Gamma$ is given by $L(\Gamma)=D(\Gamma)-A(\Gamma)$
and $N(\Gamma)=D^{\pm}(\Gamma)-A(\Gamma)$ respectively, where $D(\Gamma)$ is
the diagonal matrix with diagonal entries as vertex degree of $\Gamma$ and
$D^{\pm}(\Gamma)=$ diag $(sdeg(u_{1}),sdeg(u_{2}),\cdots,$ $sdeg(u_{m}))$ is
the net diagonal matrix of $\Gamma$. Clearly,
$L(\Gamma)-N(\Gamma)=2D^{-}(\Gamma)=2$ diag
$(d^{-}_{u_{1}},d^{-}_{u_{2}},\cdots,d^{-}_{u_{m}})$. The minimum negative
vertex degree of $\Gamma$ is given by
$\delta^{-}(\Gamma)=\text{min}\\{d^{-}_{u};u\in V(G)\\}$ and the maximum
negative vertex degree of $\Gamma$ by
$\Delta^{-}(\Gamma)=\text{max}\\{d^{-}_{u};u\in V(G)\\}$. If the signed degree
of all the vertices in $\Gamma=(G,\sigma)$ is equal to a specific integer $s$,
the graph is referred to as net-regular with a net degree of $s$ [10]. Also,
when the underlying graph $G$ is regular for some integer $r$, and $\Gamma$ is
net-regular with a net degree of $s$, the signed graph $\Gamma$ is termed co-
regular [11]. The pair $(r,s)$ is the co-regularity pair of $\Gamma$.
Furthermore, if $r=|V(\Gamma)|-1$, then $\Gamma$ is a complete co-regular
signed graph.

In any signed graph we can switch the sign of edges by a switching function
$\theta:V\rightarrow\\{+,-\\}$ [2, 8]. Switching $\Gamma=(G,\sigma)$ by
$\alpha$ create a new signed graph $\Gamma^{\alpha}=(G,\sigma^{\alpha})$ whose
underlying graph remains unchanged, but the signature is altered for each edge
$e=u_{l}u_{k}$ by $\sigma^{\alpha}(e)=\alpha(u_{l})\sigma(e)\alpha(u_{k})$.
Two signed graph $\Gamma=(G,\sigma_{1})$ and $\Sigma=(G,\sigma_{2})$ with same
underlying graph $G$ are switching equivalent (denoted as $\Gamma\sim\Sigma$)
if there exists a switching function $\alpha$ such that
$\sigma_{2}(e)=\sigma^{\alpha}_{1}(e)$ for every edge $e$ in $G$.

Lotker in [9] investigated the impact of removing a vertex on the Laplacian
spectrum in an unsigned graph. Later, Wu et al. extended the analysis in [12]
by exploring the interlacing relationship between the Laplacian spectra of the
original unsigned graph and a graph obtained by deleting more than one
vertices together with their incident edges. Also in [5], Grone et al.
demonstrated that in the case of an unsigned graph $G$, the eigenvalues of its
Laplacian matrix $L(G)$ exhibit a precise interlacing pattern with the
eigenvalues of $L(H)$ where $H$ is a graph obtained by removing a pendant
vertex along with its incident edge from $G$. Signed graph also satisfy
exactly the same interlacing relation obtained by Lotker in [9] but in section
2 we proved it using different method and use it to obtain an interlacing
relation between the Laplacian spectra of two signed cycles whose order differ
by $1$. Section 3 comprises of interlacing relation between the net-Laplacian
of signed graphs whereas in Section 4 a net-Laplacian matrix is normalised and
the interlacing relation between the normalized net-Laplacian spectra of the
signed graph $\Gamma$ and its subgraph obtained by contraction (vertex and
edge) [1] of $\Gamma$ is discussed.

Notations and results used:

  1. I.

Ordered spectrum $(\beta_{1},\beta_{2},\cdots,\beta_{n})$ refers to
eigenvalues arranged in non-decreasing order $i.e$,
$\beta_{1}\leqslant\beta_{2}\leqslant\cdots\leqslant\beta_{n}$

  2. II.

$e_{l}$ denotes a column vector with 1 in the $l^{th}$ entry and zero
elsewhere.

  3. III.

$x\perp W$ implies $x^{T}y=0$ for every $y\in W$.

  4. IV.

$\sum\limits_{\begin{subarray}{c}\Gamma(u_{l}\sim u_{t})\end{subarray}}$
indicates summation over all the unordered vertices $\\{u_{l},u_{t}\\}$ in the
graph $\Gamma$ such that $u_{l}$ and $u_{t}$ are adjacent.

  5. V.

Courant Fisher Theorem: Let $A\in\mathbb{R}^{m\times m}$ be a symmetric matrix
with ordered spectrum $(\beta_{1},\beta_{2},\cdots,\beta_{m})$. Let $p$ be any
integer such that $1\leqslant p\leqslant m$ then

$\begin{split}\beta_{p}&=\min_{W_{m-p}}\max_{{\gamma(\neq
0)\in\mathbb{R}^{m}}\atop{\gamma\perp
W_{m-p}}}\frac{\gamma^{T}A\gamma}{\gamma^{T}\gamma}\end{split}$

and

$\begin{split}\beta_{p}&=\max_{W_{p-1}}\min_{{\gamma(\neq
0)\in\mathbb{R}^{m}}\atop{\gamma\perp
W_{p-1}}}\frac{\gamma^{T}A\gamma}{\gamma^{T}\gamma}\end{split}$

where $W_{l}$ is $l$ dimensional subspace of $\mathbb{R}^{m}$ for $1\leqslant
l\leqslant m$.

  6. VI.

Weyl’s Theorem: Let $\Lambda$ and $\Theta$ be two $m\times m$ complex
Hermitian matrices and the respective ordered spectrum of $\Lambda$, $\Theta$
and $\Lambda+\Theta$ are
$(\beta_{1}(\Lambda),\beta_{2}(\Lambda),\cdots,\beta_{m}(\Lambda))$,
$(\beta_{1}(\Theta),\beta_{2}(\Theta),\cdots,\beta_{m}(\Theta))$ and
$(\beta_{1}(\Lambda+\Theta),\beta_{2}(\Lambda+\Theta),\cdots,\beta_{m}(\Lambda+\Theta))$
then for each $p=1,2,\cdots,m$ we have

$\beta_{p}(\Lambda)+\beta_{1}(\Theta)\leqslant\beta_{p}(\Lambda+\Theta)\leqslant\beta_{p}(\Lambda)+\beta_{m}(\Theta)$

The proof of Courant-Fisher theorem and Wely’s theorem is available in
[chapter 4, [7]]

##  2 Interlacing properties of eigenvalues of Laplacian Matrix

######  Theorem 1.

Suppose $\Gamma=(G,\sigma)$ is a signed graph with vertex set
$V(\Gamma)=\\{u_{1},u_{2},\cdots,u_{m+1}\\}$ and ordered Laplacian spectrum
$(\alpha_{1},\alpha_{2},\cdots,\alpha_{m+1})$ and $\Gamma^{\prime}$ is a
signed graph obtained by removing any one vertex $v$ of $\Gamma$. Let
$(\beta_{1},\beta_{2},\cdots,\beta_{m})$ be the ordered Laplacian spectrum of
$\Gamma^{\prime}$ then

$\alpha_{p}\leqslant\beta_{p}+1\leqslant\alpha_{p+1}+1~{}~{}\text{for}~{}~{}p=1,2,....,m$

###### Proof.

Suppose $v=u_{m+1}$ and $p$ be any integer such that $1\leqslant p\leqslant
m$. $\text{Let}~{}\hat{\gamma}=\begin{bmatrix}\gamma\\\ \gamma_{m+1}\\\
\end{bmatrix}$
$\in\mathbb{R}^{m+1},\gamma\in\mathbb{R}^{m},\gamma_{m+1}\in\mathbb{R}$ and
$W_{s}$ and $\hat{W}_{s}$ be $s$-dimensional subspace of $\mathbb{R}^{m}$ and
$\mathbb{R}^{m+1}$ respectively. Using Courant-Fisher Theorem V.,

$\begin{split}\alpha_{p+1}&=\min_{\hat{W}_{(m+1)-(p+1)}}~{}\max\limits_{\begin{subarray}{c}\hat{\gamma}(\neq
0)\in\mathbb{R}^{m+1}\end{subarray}\atop{\hat{\gamma}\perp\hat{W}_{(m+1)-(p+1)}}}~{}\frac{\hat{\gamma}^{T}L(\Gamma)\hat{\gamma}}{\hat{\gamma}^{T}\hat{\gamma}}\\\
&=\min_{\hat{W}_{m-p}}~{}\max\limits_{\begin{subarray}{c}\hat{\gamma}(\neq
0)\in\mathbb{R}^{m+1}\end{subarray}\atop{\hat{\gamma}\perp\hat{W}_{m-p}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}+\gamma_{m+1}^{2}}\\\
\\\
&\geqslant\min_{\hat{W}_{m-p}}~{}\max\limits_{\begin{subarray}{c}\hat{\gamma}(\neq
0)\in\mathbb{R}^{m+1}\\\
\hat{\gamma}\perp\hat{W}_{m-p}\end{subarray}\atop{\hat{\gamma}\perp
e_{m+1}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}+\sum\limits_{\begin{subarray}{c}\Gamma(u_{n+1}\sim
u_{j})\end{subarray}}(\gamma_{m+1}-\sigma(u_{m+1}u_{j})\gamma_{j})^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}+\gamma_{m+1}^{2}}\end{split}$

As $\hat{\gamma}\perp e_{m+1}$ we can consider $\gamma_{m+1}=0$ and simply
delete $(m+1)^{th}$ index of elements of $\hat{W}_{m-p}$.

$\begin{split}\therefore~{}\alpha_{p+1}&\geqslant\min_{W_{m-p}}~{}\max\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\\\ \atop{\gamma\perp
W_{m-p}}\end{subarray}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}+\sum\limits_{\begin{subarray}{c}\Gamma(u_{m+1}\sim
u_{j})\end{subarray}}\gamma_{j}^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}\\\
\\\ &\geqslant\min_{W_{m-p}}~{}\max\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\\\ \atop{\gamma\perp
W_{m-p}}\end{subarray}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}=\beta_{p}\end{split}$

and

$\begin{split}\alpha_{p}&=\max_{\hat{W}_{p-1}}~{}\min\limits_{\begin{subarray}{c}\hat{\gamma}(\neq
0)\in\mathbb{R}^{m+1}\end{subarray}\atop{\hat{\gamma}\perp\hat{W}_{p-1}}}~{}\frac{\hat{\gamma}^{T}L(\Gamma)\hat{\gamma}}{\hat{\gamma}^{T}\hat{\gamma}}\\\
&=\max_{\hat{W}_{p-1}}~{}\min\limits_{\begin{subarray}{c}\hat{\gamma}(\neq
0)\in\mathbb{R}^{m+1}\end{subarray}\atop{\hat{\gamma}\perp\hat{W}_{p-1}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{n}^{2}+\gamma_{m+1}^{2}}\\\
&\leqslant\max_{\hat{W}_{p-1}}~{}\min\limits_{\begin{subarray}{c}\hat{\gamma}(\neq
0)\in\mathbb{R}^{m+1}\\\
\hat{\gamma}\perp\hat{W}_{p-1}\end{subarray}\atop{\hat{\gamma}\perp
e_{m+1}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}+\sum\limits_{\begin{subarray}{c}\Gamma(u_{m+1}\sim
u_{j})\end{subarray}}(\gamma_{m+1}-\sigma(u_{m+1}u_{j})\gamma_{j})^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}+\gamma_{m+1}^{2}}\\\
&=\max_{W_{p-1}}~{}\min\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\\\ \atop{\gamma\perp
W_{p-1}}\end{subarray}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}+\sum\limits_{\begin{subarray}{c}\Gamma(u_{m+1}\sim
u_{j})\end{subarray}}\gamma_{j}^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}\\\
&\leqslant\max_{W_{p-1}}~{}\min\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\\\ \atop{\gamma\perp
W_{p-1}}\end{subarray}}~{}\Bigg{[}\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}+1\Bigg{]}=\beta_{p}+1\end{split}$

Thus, $\alpha_{p}\leq\beta_{p}+1\leq\alpha_{p+1}+1$ for $p=1,2,3,\cdots,m$. ∎

######  Corollary 2.

Suppose $\Gamma=(G,\sigma)$ is a signed graph of order $m+1$ whose vertex $u$
is adjacent to all the remaining vertices of $G$. If
$\Gamma^{\prime}=(G^{\prime},\sigma)$ is a signed graph obtained by removing
vertex $u$ from $\Gamma$ and $(\alpha_{1},\alpha_{2},\cdots,\alpha_{m+1})$ and
$(\beta_{1},\beta_{2},\cdots,\beta_{m})$ are respective ordered Laplacian
spectrum of $\Gamma$ and $\Gamma^{\prime}$ then,

$\alpha_{p}\leqslant\beta_{p}+1\leqslant\alpha_{p+1}~{}~{}\text{for}~{}~{}p=1,2,....,m$

###### Proof.

Let $u=u_{m+1}$. So, ${\sum\limits_{\begin{subarray}{c}\Gamma(u_{m+1}\sim
u_{j})\end{subarray}}(\gamma_{m+1}-\sigma(u_{m+1}u_{j})\gamma_{j})^{2}}={\displaystyle\sum_{j=1}^{m}(\gamma_{m+1}-\sigma(u_{m+1}u_{j})\gamma_{j})^{2}}$.
Thus

$\begin{split}\alpha_{p+1}&\geqslant\min_{W_{m-p}}~{}\max\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\\\ \atop{\gamma\perp
W_{m-p}}\end{subarray}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}+\displaystyle\sum_{j=1}^{m}\gamma_{j}^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}\\\
\\\ &=\min_{W_{m-p}}~{}\max\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\\\ \atop{\gamma\perp
W_{m-p}}\end{subarray}}~{}\Bigg{[}\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}+1\Bigg{]}=\beta_{p}+1\end{split}$

and

$\begin{split}\alpha_{p}&\leqslant\max_{W_{p-1}}~{}\min\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\\\ \atop{\gamma\perp
W_{p-1}}\end{subarray}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}+\displaystyle\sum_{j=1}^{m}\gamma_{j}^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}\\\
\\\ &=\max_{W_{p-1}}~{}\min\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\\\ \atop{\gamma\perp
W_{p-1}}\end{subarray}}~{}\Bigg{[}\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}+1\Bigg{]}=\beta_{p}+1\end{split}$

Thus, $\alpha_{p}\leqslant\beta_{p}+1\leqslant\alpha_{p+1}$ for
$p=1,2,3,\cdots,m$. ∎

######  Lemma 3.

Let $\Gamma=(G,\sigma)$ be a signed graph with $m$ vertices and Laplacian
spectrum $(\alpha_{1},\alpha_{2},\cdots\alpha_{m+1})$ and $\Gamma^{\prime}$ is
the signed graph obtained by removing any one edge $uv\in E(\Gamma)$ whose
ordered Laplacian spectrum is $(\beta_{1},\beta_{2},\cdots,\beta_{m})$ then,

$\beta_{p}\leqslant\alpha_{p}\leqslant\beta_{p}+2~{}~{}\text{for}~{}~{}k=1,2,....,m$

###### Proof.

Follows directly from Weyl’s Theorem VI.. ∎

######  Theorem 4.

Suppose $(C_{m+1},\sigma_{1})$ be a signed cycle with vertex set
$\\{v_{1},v_{2},\cdots,v_{m+1}\\}$ and ordered Laplacian spectrum
$(\alpha_{1},\alpha_{2},\cdots,\alpha_{m+1})$ and $(C_{m},\sigma_{2})$ is a
signed cycle with vertex set $\\{v_{1},v_{2},\cdots,v_{m}\\}$ and ordered
Laplacian spectrum $(\beta_{1},\beta_{2},\cdots,\beta_{m})$ with
$\sigma_{2}(e)=\sigma_{1}(e)$ for all $e\in E(C_{m})$ except for the edge
$v_{1}v_{m}$ and $\sigma_{2}(v_{1}v_{m})$ is defined randomly then,

$\alpha_{p}-1\leqslant\beta_{p}\leqslant\alpha_{p+1}+2~{}~{}\text{for}~{}~{}p=1,2,....,m$

###### Proof.

Suppose $(H,\sigma_{1})$ be a signed graph obtained by removing vertex
$v_{m+1}$ from $(C_{m+1},\sigma_{1})$ and
$(\epsilon_{1},\epsilon_{2},\cdots,\epsilon_{m})$ be the ordered Laplacian
eigenvalues of $(H,\sigma_{1})$ then by Theorem 1

$\alpha_{p}\leqslant\epsilon_{p}+1\leqslant\alpha_{p+1}+1~{}~{}~{}~{}\text{for}~{}~{}~{}~{}~{}p=1,2,\cdots,m$
(1)

Also, $(C_{m},\sigma_{2})$ is obtained by joining edge $u_{1}u_{m}$ in
$(H,\sigma_{1})$ and randomly assigning any sign to $v_{1}v_{m}$ then by Lemma
3

$\epsilon_{p}\leqslant\beta_{p}\leqslant\epsilon_{p}+2~{}~{}\text{for}~{}~{}p=1,2,\cdots,m$
(2)

Using equation 1 and equation 2 we get
$\alpha_{p}-1\leqslant\beta_{p}\leqslant\alpha_{p+1}+2~{}~{}\text{for}~{}~{}p=1,2,....,m$.
∎

######  Corollary 5.

Theorem 2.5 holds for any signature function on $C_{m+1}$ and $C_{m}$.

###### Proof.

Let L=$(C_{m+1},\sigma_{1})$ and H=$(C_{m},\sigma_{2})$, where $\sigma_{1}$
and $\sigma_{2}$ are random sign functions on $C_{m+1}$ and $C_{m}$
respectively. We can switch $L$ accordingly as it is balanced or unbalanced
shown in Figure 1.

Figure 1: Possible signed cycle of order $m+1$ upto switching isomorphism.

If H is balanced then we can remove vertex $v_{m+1}$ from L and join vertices
$v_{1}$ and $v_{m}$ with $\sigma_{1}(v_{1}v_{m})=+$ to obtain a signed graph
switching equivalent to H and if H is unbalanced then we can remove $v_{m+1}$
from L and join vertices $v_{1}$ and $v_{m}$ with $\sigma_{1}(v_{1}v_{m})=-$
to obtain a signed graph switching equivalent to H.
In both the cases result follows by Theorem 4. ∎

######  Lemma 6.

$\text{Let}~{}\gamma=\begin{bmatrix}\gamma_{1}\\\ \gamma_{2}\\\ \vdots\\\
\gamma_{m}\end{bmatrix}$ $\in\mathbb{R}^{m}$ and
$f(\gamma_{1},\gamma_{2},\cdots,\gamma_{m})$ be a bounded function on $m$
variables then for any $\gamma_{m+1}\in\mathbb{R}$

$\begin{split}\max_{\gamma\neq
0}f(\gamma_{1},\gamma_{2},\cdots,\gamma_{m})=\max_{\hat{\gamma}\neq
0}f(\gamma_{1},\gamma_{2},\cdots,\gamma_{m})~{}~{}\text{and}~{}~{}\min_{\gamma\neq
0}f(\gamma_{1},\gamma_{2},\cdots,\gamma_{m})=\min_{\hat{\gamma}\neq
0}f(\gamma_{1},\gamma_{2},\cdots,\gamma_{m})\end{split}$

where $\hat{\gamma}=\begin{bmatrix}\gamma_{1}\\\ \gamma_{2}\\\ \vdots\\\
\gamma_{m}\\\ \gamma_{m+1}\end{bmatrix}\in\mathbb{R}^{m+1}$

###### Proof.

Let function $f$ attains its maximum value at
$\alpha=(r_{1},r_{2},\cdots,r_{m})^{T}$ and minimum value at
$\beta=(t_{1},t_{2},\cdots,t_{m})^{T}$ then in $(m+1)^{th}$ dimension
$\hat{\alpha}=(r_{1},r_{2},\cdots,r_{m},a)^{T}$ gives the maximum value of $f$
for any $a\in\mathbb{R}$ and $\hat{\beta}=(t_{1},t_{2},\cdots,t_{m},b)^{T}$
gives the minimum value of $f$ for any $b\in\mathbb{R}$. ∎

######  Theorem 7.

Let $\Sigma=(G,\sigma)$ be a signed graphs of order $m+1$ with $u\in
V(\Sigma)$ such that $d(u)=1$ and $\Sigma^{\prime}=(G^{\prime},\sigma)$ be a
signed path obtained by removing vertex $u$ from $\Sigma$. Suppose $uv\in
E(\Gamma)$ with $\sigma(uv)=\star$ where $\star\in\\{+,-\\}$. If
$(\alpha_{1},\alpha_{2},\cdots,\alpha_{m+1})$ and
$(\beta_{1},\beta_{2},\cdots,\beta_{m})$ are respective ordered Laplacian
spectrum of $\Sigma$ and $\Sigma^{\prime}$ then,

$\alpha_{p}\leqslant\beta_{p}\leqslant\alpha_{p+1}~{}~{}\text{for}~{}~{}p=1,2,....,m$

###### Proof.

Taking $~{}\hat{\gamma}=\begin{bmatrix}\gamma\\\ \gamma_{m+1}\\\
\end{bmatrix}$ $\in\mathbb{R}^{m+1}$ where
$\gamma=\begin{bmatrix}\gamma_{1}\\\ \gamma_{2}\\\ \vdots\\\
\gamma_{m}\end{bmatrix}$$\in\mathbb{R}^{m},\gamma_{m+1}\in\mathbb{R}$ and
$W_{l}$, $\hat{W}_{l}$ as $l$-dimensional subspace of $\mathbb{R}^{m}$,
$\mathbb{R}^{m+1}$ respectively and proceeding as on Theorem 1 we get
$\alpha_{p+1}\geqslant\beta_{p}$ for $p=1,2,\cdots,m$. Now to prove
$\beta_{p}\geqslant\alpha_{p}$ for $p=1,2,\cdots,m$ let consider
$V(\Sigma)=\\{u_{1},u_{2},\cdots,u_{m},u_{m+1}\\}$, $u=u_{m+1}$ and $v=u_{m}$
with $u_{m+1}u_{m}\in E(\Sigma)$. Firstly we consider the case $\star$ is
negative then,

$\begin{split}\beta_{p}&=\min_{W_{m-p}}~{}\max\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\end{subarray}\atop{\gamma\perp
W_{m-p}}}~{}\frac{\gamma^{T}L(\Sigma^{\prime})\gamma}{\gamma^{T}\gamma}\\\
&=\min_{W_{m-p}}~{}\max\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\end{subarray}\atop{\gamma\perp
W_{m-p}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Sigma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}\\\
&=\min_{\hat{W}_{m-p}}~{}\max\limits_{\begin{subarray}{c}\hat{\gamma}(\neq
0)\in\mathbb{R}^{m+1}\end{subarray}\atop{\hat{\gamma}\perp\hat{W}_{m-p}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Sigma(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}-(\gamma_{m}+\gamma_{m+1})^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}[\text{by
Lemma \ref{lemma 2.6}]}\\\
&\geqslant\min_{\hat{W}_{m-p}}~{}\max\limits_{\begin{subarray}{c}\hat{\gamma}(\neq
0)\in\mathbb{R}^{m+1}\\\
\hat{\gamma}\perp\hat{W}_{m-p}\end{subarray}\atop{\gamma_{m}=-\gamma_{m+1}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Sigma(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}-(\gamma_{m}+\gamma_{m+1})^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}\\\
&=\min_{\hat{W}_{m-p}}~{}\max\limits_{\begin{subarray}{c}\hat{\gamma}(\neq
0)\in\mathbb{R}^{m+1}\\\
\hat{\gamma}\perp\hat{W}_{m-p}\end{subarray}\atop{\gamma_{m}=-\gamma_{m+1}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Sigma(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}\\\
&\geqslant\min_{\hat{W}_{m-p}}~{}\max\limits_{\begin{subarray}{c}\hat{\gamma}(\neq
0)\in\mathbb{R}^{m+1}\\\
\hat{\gamma}\perp\hat{W}_{m-p}\end{subarray}\atop{\hat{\gamma}\perp
e_{m}+e_{m+1}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Sigma(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}+\gamma_{m+1}^{2}}\\\
&=\min_{\hat{W}_{m+1-p}}~{}\max\limits_{\begin{subarray}{c}\hat{\gamma}(\neq
0)\in\mathbb{R}^{m+1}\\\
\hat{\gamma}\perp\hat{W}_{m+1-p}\end{subarray}}~{}\frac{\hat{\gamma}^{T}L(\Sigma)\hat{\gamma}}{\hat{\gamma}^{T}\hat{\gamma}}=\alpha_{p}\end{split}$

The result can be proved similarly for $\star$ as positive. ∎

######  Corollary 8.

Let $\Gamma=(P_{m+1},\sigma_{1})$ and $\Gamma^{\prime}=(P_{m},\sigma_{2})$ be
signed paths with $(m+1)$ and $m$ vertices respectively. If
$(\alpha_{1},\alpha_{2},\cdots,\alpha_{m+1})$ and
$(\beta_{1},\beta_{2},\cdots,\beta_{m})$ are respective ordered Laplacian
spectrum of $\Gamma$ and $\Gamma^{\prime}$ then,

$\alpha_{p}\leqslant\beta_{p}\leqslant\alpha_{p+1}~{}~{}\text{for}~{}~{}p=1,2,....,n$

######  Corollary 9.

Let $\Gamma=(K_{1,m},\sigma_{1})$ and $\Gamma^{\prime}=(K_{1,m-1},\sigma_{2})$
be signed stars of order $(m+1)$ and $m$ respectively. If
$(\alpha_{1},\alpha_{2},\cdots,\alpha_{m+1})$ and
$(\beta_{1},\beta_{2},\cdots,\beta_{m})$ are respective ordered Laplacian
spectrum of $\Gamma$ and $\Gamma^{\prime}$ then,

$\alpha_{p}\leqslant\beta_{p}\leqslant\alpha_{p+1}~{}~{}\text{for}~{}~{}p=1,2,....,n$

##  3 Interlacing properties of eigenvalues of net-Laplacian of signed graphs.

######  Lemma 10.

Suppose $(G,\sigma)$ is a signed graph with vertex set
{$u_{1},u_{2},\cdots,u_{m}$}. If $(\alpha_{1},\alpha_{2},\cdots,\alpha_{m})$
and $(\beta_{1},\beta_{2},\cdots,\beta_{m})$ are the respective ordered
Laplacian spectrum and net-Laplacian spectrum of $(G,\sigma)$ then for
$p=1,2,\cdots,m$

$\beta_{p}+2~{}\delta^{-}(G)\leqslant\alpha_{p}\leqslant\beta_{p}+2~{}\Delta^{-}(G)$

###### Proof.

Follows directly from Weyl’s Theorem VI.. ∎

######  Theorem 11.

Let $\Gamma=(G,\sigma)$ be a signed graph with $m$ vertices and
$\Gamma^{\prime}=\Gamma-e$ be the signed graph obtained from $\Gamma$ by
removing an edge $e$ such that $\sigma(e)=-$. If
$(\alpha_{1},\alpha_{2},\cdots,\alpha_{m})$ and
$(\beta_{1},\beta_{2},\cdots,\beta_{m})$ be the ordered net-Laplacian spectrum
of $\Gamma$ and $\Gamma^{\prime}$ respectively, then

$\alpha_{p}\leqslant\beta_{p}\leqslant\alpha_{p+1}~{}\text{ for
}~{}p=1,2,\cdots,m~{}~{}\text{ with the convention that }~{}\alpha_{m+1}=m$

###### Proof.

Suppose $p$ be any integer such that $1\leqslant p\leqslant m$.
$\text{Let}~{}{\gamma}=\begin{bmatrix}\gamma_{1}\\\ \gamma_{2}\\\ \vdots\\\
\gamma_{m}\end{bmatrix}$ $\in\mathbb{R}^{m}$, ${W}_{l}$ be $l$-dimensional
subspace of $\mathbb{R}^{m}$. Using Courant-Fisher Theorem V.,

$\begin{split}\beta_{p}&=\min_{{W}_{m-p}}~{}\max\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\end{subarray}\atop{\gamma\perp
W_{m-p}}}~{}\frac{\gamma^{T}N(\Gamma^{\prime})\gamma}{\gamma^{T}\gamma}\\\
&=\min_{W_{m-p}}~{}\max\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\end{subarray}\atop{\gamma\perp
W_{m-p}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma^{\prime})\end{subarray}}d_{i}^{-}\gamma_{i}^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}\\\
\\\ &=\min_{W_{m-p}}~{}\max\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\end{subarray}\atop{\gamma\perp
W_{m-p}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma)\end{subarray}}d_{i}^{-}\gamma_{i}^{2}+(\gamma_{1}-\gamma_{2})^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}\\\
\\\ &\geqslant\min_{W_{m-p}}~{}\max\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\end{subarray}\atop{\gamma\perp
W_{m-p}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma)\end{subarray}}d_{i}^{-}\gamma_{i}^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}=\alpha_{p}\end{split}$

and

$\begin{split}\beta_{p}&=\max_{{W}_{p-1}}~{}\min\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\end{subarray}\atop{\gamma\perp
W_{p-1}}}~{}\frac{\gamma^{T}N(\Gamma^{\prime})\gamma}{\gamma^{T}\gamma}\\\
&=\max_{W_{p-1}}~{}\min\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\end{subarray}\atop{\gamma\perp
W_{p-1}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma^{\prime})\end{subarray}}d_{i}^{-}\gamma_{i}^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}\\\
\\\ &=\max_{W_{p-1}}~{}\min\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\end{subarray}\atop{\gamma\perp
W_{p-1}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma)\end{subarray}}d_{i}^{-}\gamma_{i}^{2}+(\gamma_{1}-\gamma_{2})^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}\end{split}$
$\begin{split}\beta_{p}&\leqslant\max_{W_{p-1}}~{}\min\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\\\ \gamma\perp W_{p-1}\end{subarray}\atop{\gamma\perp
e_{1}-e_{2}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma)\end{subarray}}d_{i}^{-}\gamma_{i}^{2}+(\gamma_{1}-\gamma_{2})^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}\\\
&=\max_{W_{p}}~{}\min\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\end{subarray}\atop{\gamma\perp
W_{p}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma)\end{subarray}}d_{i}^{-}\gamma_{i}^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}=\alpha_{p+1}\end{split}$

Thus, $\alpha_{p}\leqslant\beta_{p}\leqslant\alpha_{p+1}$ for $p=1,2,\cdots,m$
∎

######  Theorem 12.

Let $\Gamma=(G,\sigma)$ be a signed graph with $m$ vertices and
$\Gamma^{\prime}=\Gamma-e$ be the signed graph obtained from $\Gamma$ by
removing an edge $e$ such that $\sigma(e)=+$. If
$(\alpha_{1},\alpha_{2},\cdots,\alpha_{m})$ and
$(\beta_{1},\beta_{2},\cdots,\beta_{m})$ be the ordered net-Laplacian
eigenvalues of $\Gamma$ and $\Gamma^{\prime}$ respectively, then

$\alpha_{p-1}\leqslant\beta_{p}\leqslant\alpha_{p}~{}\text{ for
}~{}p=1,2,\cdots,m~{}\text{ with the convention that }\alpha_{0}=-m$

###### Proof.

Similar to that of Theorem 11. ∎

######  Theorem 13.

Let $\Gamma=(G,\sigma)$ be connected signed graph with $(m+1)$ vertices and
ordered net-Laplacian eigenvalues
$(\alpha_{1},\alpha_{2},\cdots,\alpha_{m+1})$ and
$\Gamma^{\prime}=(G^{\prime},\sigma)$ is a signed graph obtained by removing
any one vertex of $\Gamma$ and $(\beta_{1},\beta_{2},\cdots,\beta_{m})$ be the
ordered net-Laplacian eigenvalues of $\Gamma^{\prime}$ then

$\alpha_{p}-1\leqslant\beta_{p}\leqslant\alpha_{p+1}+1~{}~{}\text{for}~{}~{}p=1,2,\cdots,m$

###### Proof.

Suppose $p$ be any integer such that $1\leqslant p\leqslant m$.
$\text{Let}~{}\hat{\gamma}=\begin{bmatrix}\gamma\\\ \gamma_{m+1}\\\
\end{bmatrix}$
$\in\mathbb{R}^{m+1},\gamma\in\mathbb{R}^{m},\gamma_{m+1}\in\mathbb{R}$ and
$W_{l}$ and $\hat{W}_{l}$ be $l$-dimensional subspace of $\mathbb{R}^{m}$ and
$\mathbb{R}^{m+1}$ respectively. Using Courant-Fisher Theorem V.,

$\begin{split}\alpha_{p+1}&=\min_{\hat{W}_{(m+1)-(p+1)}}~{}\max\limits_{\begin{subarray}{c}\hat{\gamma}(\neq
0)\in\mathbb{R}^{m+1}\end{subarray}\atop{\hat{\gamma}\perp\hat{W}_{(m+1)-(p+1)}}}~{}\frac{\hat{\gamma}^{T}N(\Gamma)\hat{\gamma}}{\hat{\gamma}^{T}\hat{\gamma}}\\\
&=\min_{\hat{W}_{m-p}}~{}\max\limits_{\begin{subarray}{c}\hat{\gamma}(\neq
0)\in\mathbb{R}^{m+1}\end{subarray}\atop{\hat{\gamma}\perp\hat{W}_{m-p}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma)\end{subarray}}d_{i}^{-}\gamma_{i}^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}+\gamma_{m+1}^{2}}\\\
\\\
&\geqslant\min_{\hat{W}_{m-p}}~{}\max\limits_{\begin{subarray}{c}\hat{\gamma}(\neq
0)\in\mathbb{R}^{m+1}\\\
\hat{\gamma}\perp\hat{W}_{m-p}\end{subarray}\atop{\hat{\gamma}\perp
e_{m+1}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}+\sum\limits_{\begin{subarray}{c}\Gamma(u_{m+1}\sim
u_{j})\end{subarray}}(\gamma_{m+1}-\sigma(u_{m+1}u_{j})\gamma_{j})^{2}-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma)\end{subarray}}d_{i}^{-}\gamma_{i}^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}+\gamma_{m+1}^{2}}\end{split}$

As $\hat{\gamma}\perp e_{m+1}$ we can consider $\gamma_{m+1}=0$ and simply
delete $(m+1)^{th}$ index of elements of $\hat{W}_{m-p}$.

$\begin{split}\therefore~{}\alpha_{p+1}&\geqslant\min_{W_{m-p}}~{}\max\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\\\ \atop{\gamma\perp
W_{m-p}}\end{subarray}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}+\sum\limits_{\begin{subarray}{c}\Gamma(u_{m+1}\sim
u_{j})\end{subarray}}\gamma_{j}^{2}-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma)\end{subarray}\atop{u_{i}\neq
u_{m+1}}}d_{i}^{-}\gamma_{i}^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}\\\
\end{split}$
$\begin{split}\alpha_{p+1}&\geqslant\min_{W_{m-p}}~{}\max\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\\\ \atop{\gamma\perp
W_{m-p}}\end{subarray}}~{}\left[\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma^{\prime})\end{subarray}}d_{i}^{-}\gamma_{i}^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}-1\right]=\beta_{p}-1\end{split}$

and

$\begin{split}\alpha_{p}&=\max_{\hat{W}_{p-1}}~{}\min\limits_{\begin{subarray}{c}\hat{\gamma}(\neq
0)\in\mathbb{R}^{m+1}\end{subarray}\atop{\hat{\gamma}\perp\hat{W}_{p-1}}}~{}\frac{\hat{\gamma}^{T}N(\Gamma)\hat{\gamma}}{\hat{\gamma}^{T}\hat{\gamma}}\\\
&=\max_{\hat{W}_{p-1}}~{}\min\limits_{\begin{subarray}{c}\hat{\gamma}(\neq
0)\in\mathbb{R}^{m+1}\end{subarray}\atop{\hat{\gamma}\perp\hat{W}_{p-1}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma)\end{subarray}}d_{i}^{-}\gamma_{i}^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}+\gamma_{m+1}^{2}}\\\
&\leqslant\max_{\hat{W}_{p-1}}~{}\min\limits_{\begin{subarray}{c}\hat{\gamma}(\neq
0)\in\mathbb{R}^{m+1}\\\
\hat{\gamma}\perp\hat{W}_{p-1}\end{subarray}\atop{\hat{\gamma}\perp
e_{m+1}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}+\sum\limits_{\begin{subarray}{c}\Gamma(u_{m+1}\sim
u_{j})\end{subarray}}(\gamma_{m+1}-\sigma(u_{m+1}u_{j})\gamma_{j})^{2}-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma)\end{subarray}}d_{i}^{-}\gamma_{i}^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}+\gamma_{m+1}^{2}}\\\
&=\max_{W_{p-1}}~{}\min\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\\\ \atop{\gamma\perp
W_{p-1}}\end{subarray}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}+\sum\limits_{\begin{subarray}{c}\Gamma(u_{m+1}\sim
u_{j})\end{subarray}}\gamma_{j}^{2}-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma)\end{subarray}\atop{u_{i}\neq
u_{m+1}}}d_{i}^{-}\gamma_{i}^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}\\\
&\leqslant\max_{W_{p-1}}~{}\min\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\\\ \atop{\gamma\perp
W_{p-1}}\end{subarray}}~{}\Bigg{[}\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma^{\prime})\end{subarray}}d_{i}^{-}\gamma_{i}^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}+1\Bigg{]}=\beta_{p}+1\end{split}$

Thus, $\alpha_{p}-1\leqslant\beta_{p}\leqslant\alpha_{p+1}+1$ for
$p=1,2,\cdots,m$. ∎

######  Corollary 14.

Let $\Gamma=(G,\sigma)$ be a signed graph of order $m+1$ and
$\Gamma^{\prime}=\Gamma-v$ for a vertex $v\in V(\Gamma)$ such that
$d_{v}^{-}=0$. If $(\alpha_{1},\alpha_{2},\cdots,\alpha_{m+1})$ and
$(\beta_{1},\beta_{2},\cdots,\beta_{m})$ are the ordered net-Laplacian
spectrum of $\Gamma$ and $\Gamma^{\prime}$ respectively then

$\alpha_{p}-1\leqslant\beta_{p}\leqslant\alpha_{p+1}\text{ for
}p=1,2,\cdots,m$

###### Proof.

Proof follows from Theorem 13 since $\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma)\end{subarray}\atop{u_{i}\neq
v}}d_{i}^{-}\gamma_{i}^{2}=\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma^{\prime})\end{subarray}}d_{i}^{-}\gamma_{i}^{2}$. ∎

######  Corollary 15.

Let $\Gamma=(G,\sigma)$ be a signed graph of order $m+1$ and
$\Gamma^{\prime}=\Gamma-v$ for a vertex $v\in V(\Gamma)$ such that
$d_{v}^{+}=0$. If $(\alpha_{1},\alpha_{2},\cdots,\alpha_{m+1})$ and
$(\beta_{1},\beta_{2},\cdots,\beta_{m})$ are the ordered net-Laplacian
spectrum of $\Gamma$ and $\Gamma^{\prime}$ respectively then

$\alpha_{p}\leqslant\beta_{p}\leqslant\alpha_{p+1}+1\text{ for
}p=1,2,\cdots,m$

###### Proof.

Proof follows from Theorem 13 since
$\sum\limits_{\begin{subarray}{c}\Gamma(v\sim
u_{j})\end{subarray}}\gamma_{j}^{2}-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma)\end{subarray}\atop{u_{i}\neq
v}}d_{i}^{-}\gamma_{i}^{2}=-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma^{\prime})\end{subarray}}d_{i}^{-}\gamma_{i}^{2}$. ∎

######  Corollary 16.

Let $\Gamma=(G,\sigma)$ be a $(m,k)$ co-regular signed graph of order $m+1$
where $d_{i}^{-}=s$ for all $i=1,2,\cdots,m+1$ and $\Gamma^{\prime}=\Gamma-v$
for a vertex $v\in V(\Gamma)$. If
$(\alpha_{1},\alpha_{2},\cdots,\alpha_{m+1})$ and
$(\beta_{1},\beta_{2},\cdots,\beta_{m})$ are the ordered net-Laplacian
spectrum of $\Gamma$ and $\Gamma^{\prime}$ respectively then

$\beta_{p}+2s\leqslant\alpha_{p}\leqslant\mu_{p}+(1-2s)\leqslant\alpha_{p+1}\leqslant\beta_{p+1}+2s\text{
for }p=1,2,\cdots,m$

where $(\mu_{1},\mu_{2},\cdots,\mu_{m})$ is the ordered Laplacian spectrum of
$\Gamma^{\prime}$.

###### Proof.

Let $v=u_{m+1}$. So, ${\sum\limits_{\begin{subarray}{c}\Gamma(u_{m+1}\sim
u_{j})\end{subarray}}(\gamma_{m+1}-\sigma(u_{m+1}u_{j})\gamma_{j})^{2}}={\displaystyle\sum_{j=1}^{m}(\gamma_{m+1}-\sigma(u_{m+1}u_{j})\gamma_{j})^{2}}$.
Thus

$\begin{split}\alpha_{p+1}&\geqslant\min_{W_{m-p}}~{}\max\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\\\ \atop{\gamma\perp
W_{m-p}}\end{subarray}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}+\displaystyle\sum_{j=1}^{m}\gamma_{j}^{2}-2s\sum_{i=1}^{m}\gamma_{i}^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}\\\
\\\ &=\min_{W_{m-p}}~{}\max\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\\\ \atop{\gamma\perp
W_{m-p}}\end{subarray}}~{}\left[\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}+(1-2s)\right]=\mu_{p}+(1-2s)\end{split}$

and

$\begin{split}\alpha_{p}&\leqslant\max_{W_{p-1}}~{}\min\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\\\ \atop{\gamma\perp
W_{p-1}}\end{subarray}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}+\displaystyle\sum_{j=1}^{m}\gamma_{j}^{2}-2s\sum_{i=1}^{m}\gamma_{i}^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}\\\
\\\ &=\max_{W_{p-1}}~{}\min\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\\\ \atop{\gamma\perp
W_{p-1}}\end{subarray}}~{}\left[\frac{\sum\limits_{\begin{subarray}{c}\Gamma^{\prime}(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}}{\gamma_{1}^{2}+\gamma_{2}^{2}+\cdots+\gamma_{m}^{2}}+(1-2s)\right]=\mu_{p}+(1-2s)\end{split}$

Combining the above two inequalities and using Lemma 10 will give the required
result. ∎

##  4 Normalized net-Laplacian of a signed graph

For a signed graph $\Gamma$ with vertex set $\\{u_{1},u_{2},\cdots,u_{m}\\}$,
a net-Laplacian matrix is normalized as,

$\bar{N}(\Gamma)=D(\Gamma)^{-\frac{1}{2}}N(\Gamma)D(\Gamma)^{-\frac{1}{2}}$

where,

$(D(\Gamma)^{-\frac{1}{2}})_{ij}=\begin{cases}0&\text{ if }i\neq j\\\
\frac{1}{\sqrt{d_{u_{{}_{i}}}}}&\text{ if }i=j\text{ and }d_{u_{{}_{i}}}\neq
0\\\ 0&\text{ if }i=j\text{ and }d_{u_{{}_{i}}}=0\end{cases}$

If $V(\Gamma)=\\{u_{1},u_{2},\cdots,u_{m}\\}$ and
$\gamma=\begin{bmatrix}\gamma_{1}\\\ \gamma_{2}\\\ \vdots\\\
\gamma_{m}\end{bmatrix}\in\mathbb{R}^{m}$ then,

$\frac{\gamma^{T}N(\Gamma)\gamma}{\gamma^{T}\gamma}=\frac{\sum\limits_{\begin{subarray}{c}\Gamma(v_{i}\sim
v_{j})\end{subarray}}(\gamma_{i}-\sigma(v_{i}v_{j})\gamma_{j})^{2}-2\sum\limits_{\begin{subarray}{c}v_{i}\in
V(\Gamma)\end{subarray}}d_{i}^{-}\gamma_{i}^{2}}{\displaystyle\sum_{i=1}^{n}\gamma_{i}^{2}}$

Let $y=D(\Gamma)^{\frac{1}{2}}\gamma$ then,

$\displaystyle\frac{y^{T}N(\Gamma)y}{y^{T}y}$
$\displaystyle=\frac{\left(D(\Gamma)^{\frac{1}{2}}\gamma\right)^{T}\left(D(\Gamma)^{-\frac{1}{2}}N(\Gamma)D(\Gamma)^{-\frac{1}{2}}\right)\left(D(\Gamma)^{-\frac{1}{2}}\gamma\right)}{\left(D(\Gamma)^{\frac{1}{2}}\gamma\right)^{T}\left(D(\Gamma)^{\frac{1}{2}}\gamma\right)}$
$\displaystyle=\frac{\gamma^{T}N(\Gamma)\gamma}{\gamma^{T}D\gamma}$
$\displaystyle=\frac{\sum\limits_{\begin{subarray}{c}\Gamma(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma)\end{subarray}}d_{i}^{-}\gamma_{i}^{2}}{\displaystyle\sum_{i=1}^{m}\gamma_{i}^{2}d_{i}}$
(3)

If $(\mu_{1},\mu_{2},\cdots,\mu_{m})$ is the ordered normalized net-Laplacian
spectrum of a signed graph $\Gamma=(G,\sigma)$ with vertex set
$\\{u_{1},u_{2},\cdots,u_{m}\\}$ then from [4], it follows that

$\mu_{p}\leqslant\max_{\gamma\neq
0}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma)\end{subarray}}d_{i}^{-}\gamma_{i}^{2}}{\displaystyle\sum_{i=1}^{m}\gamma_{i}^{2}d_{i}}\leqslant\max_{\gamma\neq
0}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}}{\displaystyle\sum_{i=1}^{m}\gamma_{i}^{2}d_{i}}\leqslant
2$

Equality holds for $p=m$ when $\gamma_{i}=-\sigma(u_{i}u_{j})\gamma_{j}$ for
every edge $u_{i}u_{j}$ in $G$ and $d_{i}^{-}=0$ for all $i=1,2,\cdots,m$. As
$\gamma\neq 0$, $G$ has a bipartite connected component and $\sigma=+$. Also

$\mu_{p}\geqslant\min_{\gamma\neq
0}\frac{\sum\limits_{\begin{subarray}{c}\Gamma(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma)\end{subarray}}d_{i}^{-}\gamma_{i}^{2}}{\displaystyle\sum_{i=1}^{m}\gamma_{i}^{2}d_{i}}\geqslant\min_{\gamma\neq
0}\frac{-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma)\end{subarray}}d_{i}^{-}\gamma_{i}^{2}}{\displaystyle\sum_{i=1}^{m}\gamma_{i}^{2}d_{i}}\geqslant-2$

and equality holds for $p=1$ when $\gamma_{i}=\sigma(u_{i}u_{j})\gamma_{j}$
for every edge $u_{i}u_{j}$ in $G$ and $d_{i}^{-}=d_{i}$ for all
$i=1,2,\cdots,m$. As $\gamma\neq 0$, $G$ has a bipartite connected component
and $\sigma=-$.

######  Theorem 17.

Let $\Gamma$ be a signed graph without isolated vertices of order $m$ and let
$\Gamma^{\prime}=\Gamma-e$ be the signed graph obtained by removing an edge
$e=uv$ such that $\sigma(e)=-$. If $(\alpha_{1},\alpha_{2},\cdots,\alpha_{m})$
and $(\beta_{1},\beta_{2},\cdots,\beta_{m})$ are the ordered spectrum of
$\bar{N}(\Gamma)$ and $\bar{N}(\Gamma^{\prime})$ respectively, then

$\alpha_{p}\leqslant\beta_{p}\leqslant\alpha_{p+2}~{};~{}\text{ for
}p=1,2,\cdots,m$

###### Proof.

Taking $v=u_{1}$ and $u=u_{2}$ and proceeding as in Theorem 11, the
eigenvalues $\beta_{p}$ of $\bar{N}(\Gamma^{\prime})$ can be expressed as

$\beta_{p}=\min_{W_{m-p}}~{}\max\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\end{subarray}\atop{\gamma\perp
W_{m-p}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma)\end{subarray}}d_{i}^{-}\gamma_{i}^{2}+(\gamma_{1}-\gamma_{2})^{2}}{\sum\limits_{\begin{subarray}{c}\Gamma:i=1\end{subarray}}^{m}d_{i}\gamma_{i}^{2}-\gamma_{1}^{2}-\gamma_{2}^{2}}$

Similar to Theorem 11, the lower bound $\alpha_{p}$ follows and the upper
bound $\alpha_{p+2}$ follows taking $\gamma\perp e_{1},e_{2}$ in the max-min
statement of Courant-Fisher Theorem V.. ∎

######  Theorem 18.

Let $\Gamma$ be a signed graph without isolated vertices of order $m$ and let
$\Gamma^{\prime}=\Gamma-e$ be the signed graph obtained by removing an edge
$e=uv$ such that $\sigma(e)=+$. If $(\alpha_{1},\alpha_{2},\cdots,\alpha_{m})$
and $(\beta_{1},\beta_{2},\cdots,\beta_{m})$ are the ordered spectrum of
$\bar{N}(\Gamma)$ and $\bar{N}(\Gamma^{\prime})$ respectively, then

$\alpha_{p-1}\leqslant\beta_{p}\leqslant\alpha_{p+1}~{};~{}\text{ for
}p=1,2,\cdots,m$

###### Proof.

Taking $v=u_{1}$ and $u=u_{2}$ and proceeding as in Theorem 11, the
eigenvalues $\beta_{p}$ of $\bar{N}(\Gamma^{\prime})$ can be expressed as

$\beta_{p}=\min_{W_{m-p}}~{}\max\limits_{\begin{subarray}{c}\gamma(\neq
0)\in\mathbb{R}^{m}\end{subarray}\atop{\gamma\perp
W_{m-p}}}~{}\frac{\sum\limits_{\begin{subarray}{c}\Gamma(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}-2\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma)\end{subarray}}d_{i}^{-}\gamma_{i}^{2}-(\gamma_{1}-\gamma_{2})^{2}}{\sum\limits_{\begin{subarray}{c}\Gamma:i=1\end{subarray}}^{m}d_{i}\gamma_{i}^{2}-\gamma_{1}^{2}-\gamma_{2}^{2}}$

Similar to Theorem 8 in [1], taking $\gamma_{1}=\gamma_{2}$ gives the lower
bound $\alpha_{p-1}$ and the upper bound $\alpha_{p+1}$ follows taking
$\gamma_{1}=-\gamma_{2}$ in the max-min statement of Courant-Fisher Theorem
V.. ∎

###  4.1 Vertex contraction

Consider a signed graph $\Gamma$, and let $t\in V(\Gamma)$. The open
neighborhood of the vertex $t$ is defined as the set of vertices adjacent to
$t$ in $\Gamma$ that is

$\mathcal{N}(t)=\\{w\in V(\Gamma);~{}~{}tw\in E(\Gamma)\\}$

and $\mathcal{N}[t]=\mathcal{N}(t)\cup\\{t\\}$ is a closed neighborhood of $t$
in $\Gamma$.

For a signed graph $\Gamma=(G,\sigma)$ with vertices $a$ and $b$ such that
$\sigma(ta)=\sigma(tb)$ for all $t\in\mathcal{N}(a)\cap\mathcal{N}(b)$, Atay
and Tuncel in [1] defined an allowable contraction $\Gamma/\\{a,b\\}$ on $a$
and $b$, as a signed graph obtained from $\Gamma$ by deleting the vertices $a$
and $b$ and adding a new vertex $(ab)$ such that
$\mathcal{N}(ab)=\mathcal{N}(a)\cup\mathcal{N}(b)$ and

$\sigma(t(ab))=\begin{cases}\sigma(ta)&\text{ if }t\in\mathcal{N}(a)\text{ and
}t\notin\mathcal{N}(b)\\\ \sigma(tb)&\text{ if }t\in\mathcal{N}(b)\text{ and
}t\notin\mathcal{N}(a)\\\ \sigma(ta)=\sigma(tb)&\text{ if
}t\in\mathcal{N}(a)\cap\mathcal{N}(b)\end{cases}$

In particular, if $\mathcal{N}(a)\cup\mathcal{N}[b]=\emptyset$ then
$\Gamma/\\{a,b\\}$ is an allowable contraction.

######  Theorem 19.

Let $\Gamma$ be a signed graph and $a,b\in V(\Gamma)$ such that
$\mathcal{N}(a)\cap\mathcal{N}(b)=\emptyset$. If
$(\alpha_{1},\alpha_{2},\cdots,\alpha_{m+1})$ and
$(\beta_{1},\beta_{2},\cdots,\beta_{m})$ are the ordered spectrum of
$\bar{N}(\Gamma)$ and $\bar{N}(\Gamma^{\prime})$ respectively where
$\Gamma^{\prime}=\Gamma/\\{a,b\\}$, then for $p=1,2,\cdots,m$

$\alpha_{p-1}\leqslant\beta_{p}\leqslant\alpha_{p+1}$

with the convention that $\alpha_{0}=-2$

###### Proof.

Let $a=u_{1}$, $b=u_{2}$ and $k$ denote an index set such that $k\in K$ if and
only if $u_{k}\in N(u_{1})$. As,
$\mathcal{N}(u_{1})\cap\mathcal{N}(u_{2})=\emptyset$, $\Gamma/\\{a,b\\}$ can
be obtained from $\Gamma$ by removing edges $u_{1}u_{k}$ and adding edges
$u_{2}u_{k}$ such that $\sigma(u_{2}u_{k})=\sigma(u_{1}u_{k})$ for all $k\in
K$. Using Courant-Fisher Theorem V., the eigenvalues $\beta_{p}$ of
$\mathcal{N}(\Gamma^{\prime})$ is given by

$\begin{split}\beta_{p}&=\min_{W_{m-p}}~{}\max\limits_{\begin{subarray}{c}\gamma\neq
0\end{subarray}\atop{\gamma\perp
W_{m-p}}}~{}\frac{1}{\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma)\end{subarray}}d_{i}\gamma_{i}^{2}-\gamma_{1}^{2}-\gamma_{2}^{2}}\Bigg{[}\sum\limits_{\begin{subarray}{c}\Gamma(u_{i}\sim
u_{j})\end{subarray}}(\gamma_{i}-\sigma(u_{i}u_{j})\gamma_{j})^{2}+\sum\limits_{\begin{subarray}{c}k\in
K\end{subarray}}\Big{[}(\gamma_{2}-\sigma(u_{2}u_{k})\gamma_{k})^{2}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-(\gamma_{1}-\sigma(u_{1}u_{k})\gamma_{k})^{2}\Big{]}-2\Big{(}\sum\limits_{\begin{subarray}{c}u_{i}\in
V(\Gamma)\end{subarray}}d_{i}^{-}\gamma_{i}^{2}-d_{1}^{-}\gamma_{1}^{2}+d_{1}^{-}\gamma_{2}^{2}\Big{)}\Bigg{]}\end{split}$

Similar to Theorem 2.7 in [3], taking $\gamma_{1}=\gamma_{2}$ on both min-max
and max-min statement of the Courant-Fisher Theorem V. gives the lower bound
$\alpha_{p-1}$ and upper bound $\alpha_{p+1}$ respectively. ∎

## Acknowledgements

We would like to acknowledge National Institute of Technology Sikkim for
giving doctoral fellowship to Satyam Guragain.

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