#  Bounds and extremal graphs for the energy of complex unit gain
graphs111Some of the results of this article are part of the PhD thesis of
first author

Aniruddha Samanta Theoretical Statistics and Mathematics Unit, Indian
Statistical Institute, Kolkata-700108, India. Email<EMAIL_ADDRESS>M. Rajesh Kannan Department of Mathematics, Indian Institute of Technology
Hyderabad, Hyderabad 502285, India. Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A complex unit gain graph ($\mathbb{T}$-gain graph), $\Phi=(G,\varphi)$ is a
graph where the gain function $\varphi$ assigns a unit complex number to each
orientation of an edge of $G$ and its inverse is assigned to the opposite
orientation. The associated adjacency matrix $A(\Phi)$ is defined canonically.
The energy $\mathcal{E}(\Phi)$ of a $\mathbb{T}$-gain graph $\Phi$ is the sum
of the absolute values of all eigenvalues of $A(\Phi)$. For any connected
triangle-free $\mathbb{T}$-gain graph $\Phi$ with the minimum vertex degree
$\delta$, we establish a lower bound $\mathcal{E}(\Phi)\geq 2\delta$ and
characterize the equality. Then, we present a relationship between the
characteristic and the matching polynomial of $\Phi$. Using this, we obtain an
upper bound for the energy $\mathcal{E}(\Phi)\leq 2\mu\sqrt{2\Delta_{e}+1}$
and characterize the classes of graphs for which the bound sharp, where $\mu$
and $\Delta_{e}$ are the matching number and the maximum edge degree of
$\Phi$, respectively. Further, for any unicyclic graph $G$, we study the gains
for which the gain energy $\mathcal{E}(\Phi)$ attains the maximum/minimum
among all $\mathbb{T}$-gain graphs defined on $G$.

AMS Subject Classification(2010): 05C50, 05C22, 05C35.

Keywords. Gain energy, Energy of a vertex, Matching polynomial, Maximum edge
degree, Coulsion integral formula.

##  1 Introduction

Let $G$ be a simple graph with vertex set $V(G)=\\{v_{1},v_{2},\dots,v_{n}\\}$
and edge set $E(G)$. We denote $v_{s}\sim v_{t}$ if the vertices $v_{s}$ and
$v_{t}$ are adjacent, and the edge between them is denoted by $e_{s,t}$. The
_adjacency matrix_ $A(G)$ of a graph $G$ is a symmetric $n\times n$ matrix
whose $(s,t)th$ entry is $1$ if $v_{s}\sim v_{t}$ and zero otherwise. Let the
multiset $\\{\lambda_{1},\dots,\lambda_{n}\\}$ be the eigenvalues (or the
spectrum) of $A(G)$. The _energy_ of the graph $G$ is defined as

$\mathcal{E}(G)=\sum\limits_{k=1}^{n}|\lambda_{k}|.$

Graph energy was systematically studied by Gutman [11] in 1978, which was
motivated by the notion of total $\pi$-electron energy from chemistry. Graph
energy has several applications in science and engineering. For a detailed
study on graph energy and its applications, we refer to [14]. This paper aims
to study the energy of $\mathbb{T}$-gain graphs. A _directed graph(or
digraph)_ $X$ is an order pair $(V(X),E(X))$,
where$V(X)=\\{v_{1},v_{2},\dots,v_{n}\\}$ is the vertex set and $E(X)$ is the
directed edge set. A directed edge from the vertex $v_{p}$ to the vertex
$v_{q}$ is denoted by $\vec{e}_{p,q}$.

From a simple graph $G$, by orienting each undirected edge $e_{p,q}\in E(G)$
in two opposite directions, namely $\vec{e}_{p,q}$ and $\vec{e}_{q,p}$, we get
a digraph. Let $\vec{E}(G)=\\{\vec{e}_{p,q},\vec{e}_{q,p}:e_{p,q}\in E(G)\\}$
and $\mathbb{T}=\\{z\in\mathbb{C}:|z|=1\\}$. A _complex unit gain graph_
(simply, $\mathbb{T}$-gain graph) on a simple graph $G$ is a pair
$(G,\varphi)$, where $\varphi:\vec{E}(G)\rightarrow\mathbb{T}$ is a mapping
such that $\varphi(\vec{e}_{p,q})=\varphi(\vec{e}_{q,p})^{-1}$. A
$\mathbb{T}$-gain graph $(G,\varphi)$ is denoted by $\Phi$. For more details
about the $\mathbb{T}$-gain graphs, we refer to [5, 6, 23, 24, 30, 31].

The _adjacency matrix_ of $\Phi$ is the Hermitian matrix
$A(\Phi)=(a_{p,q})_{n\times n}$ defined as follows:

$a_{p,q}=\begin{cases}\varphi(\vec{e}_{p,q})&\text{if }\mbox{$v_{p}\sim
v_{q}$},\\\ 0&\text{otherwise.}\end{cases}$

Let the multiset $\\{\lambda_{1},\dots,\lambda_{n}\\}$ be the eigenvalues of
$A(\Phi)$ (or the spectrum of $\Phi$), and is denoted by
$\operatorname{spec}(\Phi)$. The energy of $\Phi$, denoted by
$\mathcal{E}(\Phi)$, is defined by $\sum\limits_{j=1}^{n}|\lambda_{j}|$. The
spectral radius of $\Phi$, denoted by $\rho(\Phi)$, is the maximum of absolute
values of all the eigenvalues of $\Phi$. The adjacency matrices of
$\mathbb{T}$-gain graphs include several known classes of adjacency: Signed
adjacency matrices [32], Hermitian adjacency matrices are mixed graphs of
first kind and second kind [10, 15, 19], complex adjacency matrices with
entries are from the set $\\{\pm 1,\pm i\\}$ [4] and so on.

The _degree_ of a vertex $v$ of a graph $G$ is the number of edges incident
with $v$, denoted by $d(v)$. Then, the maximum and the minimum vertex degree
of $G$ are denoted by $\Delta(G)$ and $\delta(G)$, respectively. In [16], Ma
proved the following bound for the energy of a simple graph in terms of the
minimum vertex degree.

######  Theorem 1.1 ([16, Theorem 1.2]).

Let $G$ be a connected simple graph with the minimum vertex degree $\delta$.
Then $\mathcal{E}(G)\geq 2\delta$ and equality occurs if and only if $G$ is a
multipartite graph with equal partition size.

Later, Oboudi [21] gave a simple proof of the above result. Our first
objective in this article is to establish a lower bound for the energy of a
connected triangle-free $\mathbb{T}$-gain graph in terms of the minimum vertex
degree and to characterize the sharpness of the inequality with the help of
vertex energy. First, we show that the counterpart of Theorem 1.1 need not be
true for $\mathbb{T}$-gain graphs (See, example 3.1). Surprisingly, the
counterpart of Theorem 1.1 holds for triangle-free $\mathbb{T}$-gain graphs.
Moreover, we characterize the $\mathbb{T}$-gain graphs for the bound is sharp.
This is done in Section 3.

The _degree of an edge $e$_ of $G$ is the number of edges incident with it.
Let $\Delta_{e}(G)$ denote the maximum edge degree of $G$. A _matching_ in a
simple graph $G$ is a set of independent edges of $G$. The _matching number_
$\mu(G)$ is the cardinality of a maximum matching of $G$. Wang and Ma [27]
established a lower bound $\mathcal{E}(G)\geq 2\mu(G)$ for bipartite graph
$G$. In [29], Dein Wong et al. proved the result for any simple graph $G$.
Later, this result has been extended for mixed graphs [28] and
$\mathbb{T}$-gain graphs [24]. Yingui Pan et al. [22] obtained the following
upper bound for energy $\mathcal{E}(G)$ in terms of $\mu(G)$. Let $P_{n}$
denote the path graph on $n$ vertices.

######  Theorem 1.2 ([22, Theorem 1]).

Let $G$ be a simple graph with the matching number $\mu$ and maximum edge
degree $\Delta_{e}$. Then

  * (i)

If $\Delta_{e}$ is even, then $\mathcal{E}(G)\leq 2\mu\sqrt{2\Delta_{e}+1}$.
Equality occurs if and only if $G$ is the disjoint union of $\mu$ copies of
$P_{2}$ and some isolated vertices.

  * (ii)

If $\Delta_{e}$ is odd, then
$\mathcal{E}(G)\leq\mu\left(\sqrt{b+2\sqrt{b}}+\sqrt{b-2\sqrt{b}}\right)$,
where $b=2(\Delta_{e}+1)$. Equality occurs if and only if $G$ is $\mu$ copies
of $P_{3}$ and some isolated vertices.

In [2], Akbari et al. obtained an alternative simple proof of Theorem 1.2.
Note that this result was first proved by Tian and Wong [26] for triangle-free
graphs. Our second objective of this article is to extend the above bound for
the $\mathbb{T}$-gain graphs. The main idea of the proof of Akbari et al.
depends on graph decomposition. This decomposition technique is not helpful
for the $\mathbb{T}$-gain graphs (see Example 4.1). We develop an alternate
graph decomposition technique for the $\mathbb{T}$-gain graphs. Also, we
establish an integral representation for the $\mathbb{T}$-gain graphs and a
relation between the characteristic and matching polynomials of
$\mathbb{T}$-gain graphs. Using these ideas, we extend Theorem 1.2 for the
$\mathbb{T}$-gain graphs. This is done in Section 4.

Let $\mathcal{T}_{G}$ denote the collection of all $\mathbb{T}$-gain graphs
defined on $G$. A natural question that arises here is the following: For
which simple graph $G$, the energy $\mathcal{E}(G)$ is minimum/maximum among
its $\mathbb{T}$-gain graphs $\mathcal{T}_{G}$ ?

For a graph $G$, the Seidel adjacency matrix is defined as follows:
$S(G)=J-2A(G)-I$, where $J$ is the all one’s matrix and $A(G)$ is the
adjacency matrix of $G$. The Seidel adjacency matrix of any graph $G$ can be
viewed as a signed adjacency matrix of a complete signed graph. One of the
conjectures regarding the Seidel energy given by Haemers [12] can be rephrased
as follows: $\mathcal{E}(K_{n})\leq\mathcal{E}(K_{n},\psi)$, for any signed
graph $(K_{n},\psi)$. Akbari et al. affirmatively settled this conjecture [1].
In Section 5, first, we show that the Haemers conjecture need not be true for
a $\mathbb{T}$-gain graph $(K_{n},\psi)$. Moreover, in the case of
$\mathbb{T}$-gain graphs, the extremal energy need not be attained for the
graph $G$ (unlike the signed case). So it is natural to ask the following
question: Identify the classes of graphs for which the extremal energy is the
energy of $G$. In [24], we established that
$\mathcal{E}(K_{n,n})\leq\mathcal{E}(K_{n,n},\varphi)$ for any
$\mathbb{T}$-gain graph $(K_{n,n},\varphi)$. In Theorem 5.1, we show that for
any unicyclic graph $G$, $\mathcal{E}(G)$ attains extremum(either maximum or
minimum) energy among all the $\mathbb{T}$-gain graphs $\Phi=(G,\varphi)$. The
extremal families are also characterized. Two $\mathbb{T}$-gain graphs
$\Phi_{1}$ and $\Phi_{2}$ are  equienergetic if
$\mathcal{E}(\Phi_{1})=\mathcal{E}(\Phi_{2})$. If $G$ is a tree, then any two
elements of $\mathcal{T}_{G}$ are equienergetic. In Theorem 5.2, we show that
if $G$ is a graph such that all its cycles are vertex disjoint, then
$\mathcal{T}_{G}$ is not equienergetic. Finally, we conjecture that
$\mathcal{T}_{G}$ is equienergetic if and only if $G$ is a tree.

##  2 Preliminaries

This section recalls some known definitions and results necessary for this
article. Let $G$ be an undirected simple graph. If $G$ contains no triangle as
an induced subgraph, it is called a _triangle-free_ graph. Let us recall the
following couple of results for undirected graphs.

######  Theorem 2.1 ([13, Theorem 4.7]).

(Mantel’s theorem) Any $n$-vertices triangle-free graph has at most
$\lfloor\frac{n^{2}}{4}\rfloor$ edges.

######  Theorem 2.2 ([25, Theorem 2.38]).

Let $G$ be a graph of $m$ edges with the largest eigenvalue $\lambda_{1}(G)$.
Then $\lambda_{1}(G)\leq\sqrt{m}.$

A _clique_ of a graph $G$ is a complete subgraph. The _clique number_ of a
graph $G$ is the number of vertices of a clique with maximum size, and it is
denoted by $\omega(G)$. The following result is due to Nikiforov [20].

######  Theorem 2.3 ([20, Theorem 2]).

Let $G$ be a graph with no isolated vertices. Let $\lambda_{1}(G),m,$ and
$\omega(G)$ be the largest eigenvalue, number of edges, and clique number of
$G$, respectively. Then

$\lambda_{1}(G)^{2}\leq\frac{2(\omega(G)-1)}{\omega(G)}m.$

Equality holds if and only if one of the following occurs:

  * (i)

$\omega(G)=2$ and $G$ is complete bipartite graph.

  * (ii)

$\omega(G)\geq 3$ and $G$ is a complete regular $\omega(G)$-partite graph.

Let $G$ be a simple connected graph and $v_{i},v_{j}$ be any two vertices of
$G$. Then the _distance_ between $v_{i}$ and $v_{j}$ is the length of the
shortest path in between $v_{i}$ and $v_{j}$. The _diameter_ of a graph $G$ is
the maximum distance between any two vertices in $G$. The _diameter_ of a
$\mathbb{T}$-gain graph is the diameter of its underlying graph.

Figure 1:  Graph $T_{1}$

######  Lemma 2.1 ([26, Lemma 2.1]).

Let $\mathcal{A}_{\Delta_{e},3}$ be the set of all diameter $3$ trees with the
maximum edge degree $\Delta_{e}$. Then for any
$T\in\mathcal{A}_{\Delta_{e},3}$, $\mathcal{E}(T)\leq\mathcal{E}(T_{1})$,
where $T_{1}\in\mathcal{A}_{\Delta_{e},3}$ is shown in Figure 1. Equality
occurs if and only if $T=T_{1}$.

######  Lemma 2.2 ([26, Lemma 2.3]).

Let $T_{1}$ be a tree given in Figure 1. Then
$\mathcal{E}(T_{1})=2\sqrt{2\Delta_{e}+1}$ if $\Delta_{e}$ is even and
$\mathcal{E}(T_{1})=\sqrt{b+2\sqrt{b}}+\sqrt{b-2\sqrt{b}}$ if $\Delta_{e}$ is
odd, where $b=2(\Delta_{e}+1)$.

The maximum vertex degree, minimum edge degree, maximum edge degree and
matching number of a $\mathbb{T}$-gain graph $\Phi=(G,\varphi)$ are defined as
that of the underlying graph $G$. Two $\mathbb{T}$-gain graphs
$\Phi_{1}=(G,\varphi_{1})$ and $\Phi_{2}=(G,\varphi_{2})$ are _switching
equivalent_ if there exist a unitary diagonal matrix $U$ such that
$A(\Phi_{2})=UA(\Phi_{1})U^{*}$. We write $\Phi_{1}\sim\Phi_{2}$, if
$\Phi_{1}$ and $\Phi_{2}$ are switching equivalent. If $\Phi_{1}\sim\Phi_{2}$
then $\operatorname{spec}(\Phi_{1})=\operatorname{spec}(\Phi_{2})$. Let $C$ be
a cycle of $n$ vertices, and the vertices are arranged as follows:
$C:v_{1}\sim v_{2}\sim\cdots\sim v_{n}\sim v_{1}$. Now the gain of $\vec{C}$
is defined as
$\varphi(\vec{C})=\varphi(\vec{e}_{1,2})\varphi(\vec{e}_{2,3})\dots\varphi(\vec{e}_{n,n-1})\varphi(\vec{e}_{n,1})$.
For a complex number $\lambda$, let $\operatorname{Re}(\lambda)$ denote the
real part of $\lambda$. For any cycle $C$, instead of
$\operatorname{Re}(\varphi(\vec{C}))$, we simply write
$\operatorname{Re}(\varphi(C))$. A $\mathbb{T}$-gain graph $\Phi=(G,\varphi)$
is called _balanced_ if $\varphi(C)=1$, for all cycle $C$ in $G$. If $\Phi$ is
balanced, then $\Phi\sim(G,1)$.

Now, we recall a couple of results for the spectrum of $\mathbb{T}$-gain
graphs.

######  Lemma 2.3 ([18, Corollary 3.1]).

Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph with characteristic
polynomial
$P_{\Phi}(x)=x^{n}+b_{1}(\Phi)x^{n-1}+b_{2}(\Phi)x^{n-2}+\cdots+b_{n}(\Phi)$.
Then

$\displaystyle
b_{i}(\Phi)=\sum\limits_{H\in\mathcal{H}_{i}(G)}(-1)^{n(H)}2^{c(H)}\prod\limits_{C\in\mathcal{C}(H)}\operatorname{Re}(\varphi(C)),~{}~{}\text{for}~{}i=1,2,\dots,n$
(1)

where $\mathcal{H}_{i}(G)$ is the set of all elementary subgraphs with $i$
vertices in $G$ and $n(H),~{}c(H),~{}\mathcal{C}(H)$ are the number of
components, number of cycles and collection of cycles in $H$, respectively.

######  Theorem 2.4 ([18, Lemma 4.1, Theorem 4.4]).

Let $\Phi=(G,\varphi)$ be any $\mathbb{T}$-gain graph on a connected graph
$G$. Then $\rho(\Phi)\leq\rho(G)$. Equality occurs if and only if either
$\Phi$ or $-\Phi$ is balanced.

######  Theorem 2.5 ([18, Theorem 4.6]).

Let $\Phi$ be a $\mathbb{T}$-gain graph on an underlying graph $G$. Then
$\operatorname{spec}(\Phi)=\operatorname{spec}(G)$ if and only if $\Phi$ is
balanced.

######  Theorem 2.6 ([18, Theorem 4.1]).

Let $\Phi$ be a bipartite $\mathbb{T}$-gain graph. Then, the spectrum of
$\Phi$ is symmetric about the origin.

######  Theorem 2.7 ([18, Theorem 4.5]).

If $\Phi=(G,\varphi)$ be a connected $\mathbb{T}$-gain graph, then

  * (1)

If $G$ is bipartite, then $\Phi$ is balanced implies $-\Phi$ is balanced.

  * (2)

If $\Phi$ is balanced implies $-\Phi$ is balanced for some gain, then $G$ is
bipartite.

######  Theorem 2.8 ([24, Theorem 4.2]).

Let $\Phi=(G,\varphi)$ be any $\mathbb{T}$-gain graph on a connected bipartite
graph $G$. Then $\Phi$ has exactly one positive eigenvalue if and only if
$\Phi$ is a balanced complete bipartite graph.

Arizmendi et al. [3] introduced the notion of the energy of a vertex for an
undirected graph. Later, in [24], we extended this notion for
$\mathbb{T}$-gain graphs. Let $v_{i}\in V(\Phi)$. Then the _energy of the
vertex_ $v_{i}$ in $\Phi$ is defined by
$\mathcal{E}_{\Phi}(v_{i}):=|A(\Phi)|_{ii}$, where $|A(\Phi)|_{ii}$ is the
$(i,i)th$ entry of the matrix $(A(\Phi)A(\Phi)^{*})^{\frac{1}{2}}$. Note that,
$\mathcal{E}(\Phi)=\sum\limits_{i=1}^{n}\mathcal{E}_{\Phi}(v_{i})$.

######  Lemma 2.4 ([24, Lemma 3.1]).

Let $\Phi$ be a $\mathbb{T}$-gain graph with vertex set
$\\{v_{1},v_{2},\cdots,v_{n}\\}$. Then

$\mathcal{E}_{\Phi}(v_{i})=\sum\limits_{j=1}^{n}P_{ij}|\lambda_{j}|,\text{ for
}i=1,2,\dots n,$

where $P_{ij}=|q_{ij}|^{2}$ and $Q=(q_{ij})$ is the unitary matrix whose
columns are the eigenvectors of $A(\Phi)$ and $\lambda_{j}$ is the $j$-th
eigenvalue of $A(\Phi)$.

The following integral representation will be useful.

######  Theorem 2.9 ([17, Theorem 1]).

Let $X(t)$ be a polynomial of degree $n$ with leading coefficient $1$ and
$\lambda_{1},\lambda_{2},\dots,\lambda_{n}$ be the roots of $X(t)$. Then

$\displaystyle\sum\limits_{j}|\operatorname{Re}(\lambda_{j})|=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{1}{t^{2}}\log\left|t^{n}X\left(\frac{i}{t}\right)\right|dt.$
(2)

Next, we recall a couple of results from matrix theory, which will be used in
the proofs of the main results.

######  Theorem 2.10 ([8, Corollary 2.4]).

Let $X=\begin{pmatrix}Y&A\\\ B&Z\end{pmatrix}$ be a partition matrix such that
$Y$ and $Z$ are the square matrices. Then $\mathcal{E}(X)\geq\mathcal{E}(Y)$.
Equality holds if and only if $A,B$, and $Z$ are zero matrices.

######  Theorem 2.11 ([7]).

Let $C,~{}C_{1}$ and $C_{2}$ be three square complex matrices of order $n$
such that $C=C_{1}+C_{2}$. If $S_{j}(\cdot{})$ is the $j$-th singular value of
corresponding matrix, then
$\sum\limits_{p}S_{p}(C)\leq\sum\limits_{p}S_{p}(C_{1})+\sum\limits_{p}S_{p}(C_{2})$.

##  3 Lower bounds for energy of gain graphs in terms of minimum vertex degree

In [16], Ma established that $\mathcal{E}(G)\geq 2\delta(G)$ for a connected
undirected graph $G$ and studied the classes of graphs for which the equality
hold, where $\delta(G)$ is the minimum vertex degree. First, we note that this
bound need not be true for arbitrary connected $\mathbb{T}$-gain graphs.

######  Example 3.1.

Consider $\Phi=(K_{3},\varphi)$, where $K_{3}$ denotes the complete graph on
$3$ vertices, with the gains are given by
$\varphi(\vec{e}_{1,2})=\varphi(\vec{e}_{2,3})=\varphi(\vec{e}_{3,1})=i$. Then
$A(\Phi)=\begin{pmatrix}0&i&-i\\\ -i&0&i\\\ i&-i&0\end{pmatrix}$. Then
$\operatorname{spec}(\Phi)=\\{-\sqrt{3},0,\sqrt{3}\\}$. Thus
$\mathcal{E}(\Phi)=2\sqrt{3}\approx 3.464<2\delta(\Phi)$.

Surprisingly, the counterpart of the result of Ma holds for the triangle-free
connected $\mathbb{T}$-gain graphs, which is the main result of this section
(Theorem 3.3). We shall prove this result with the aid of the following
auxiliary results. For a gain graph $\Phi=(G,\varphi)$, we first establish a
lower bound for the energy of a vertex in terms of the vertex degrees and the
spectral radius of the graph $G$.

######  Theorem 3.1.

Let $\Phi=(G,\varphi)$ be a connected $\mathbb{T}$-gain graph with at least
one edge. Then

$\mathcal{E}_{\Phi}(v_{i})\geq\frac{d(v_{i})}{\rho(G)},\qquad\text{for all
}v_{i}\in V(G).$ (3)

Further, equality holds in (3) if and only if $\Phi$ is balanced and $G$ is
complete bipartite.

###### Proof.

Let $\lambda_{n}\leq\lambda_{n-1}\leq\cdots\leq\lambda_{1}$ be the eigenvalues
of $\Phi$. Then, by Theorem 2.4, $\rho(\Phi)\leq\rho(G)$, where
$\rho(\Phi)=\max\\{-\lambda_{n},\lambda_{1}\\}$. Therefore
$-\rho(G)\leq\lambda_{i}\leq\rho(G)$ for all $i$. Since
$\left|\frac{\lambda_{i}}{\rho(G)}\right|\leq 1$, so
$\left(\frac{\lambda_{i}}{\rho(G)}\right)^{2}\leq\left|\frac{\lambda_{i}}{\rho(G)}\right|$;
and equality holds if and only if $\lambda_{i}\in\\{-\rho(G),0,\rho(G)\\}$.
Note that, the $i$-th diagonal entry of $A(\Phi)^{2}$ is $d(v_{i})$. Thus, by
Lemma 2.4, we get

$\mathcal{E}_{\Phi}(v_{i})=\sum\limits_{j=1}^{n}|q_{ij}|^{2}|\lambda_{j}|\geq\sum\limits_{j=1}^{n}|q_{ij}|^{2}\frac{\lambda_{j}^{2}}{\rho(G)}=\frac{d(v_{i})}{\rho(G)},$
(4)

where $Q=(q_{ij})$ is the unitary matrix whose columns are eigenvectors of
$\Phi$.

For any $i$, equality holds in the equation (4) if and only if
$\lambda_{j}\in\\{-\rho(G),0,\rho(G)\\}$ for all $j$. As $G$ has at least one
edge, a non-zero eigenvalue exists, say $\lambda_{i}$, different from zero.
Thus $|\lambda_{i}|=\rho(G)$, and hence $\rho(\Phi)=\rho(G)$. Therefore, by
Theorem 2.4, either $\Phi$ is balanced or $-\Phi$ is balanced.

If $\Phi$ is balanced, then, by Theorem 2.5,
$\operatorname{spec}(\Phi)=\operatorname{spec}(G)$. As $G$ is connected by the
Perron-Frobenius theorem, $\rho(G)$ is a simple eigenvalue of $A(G)$. Since
the trace of $A(G)$ is zero, so
$\operatorname{spec}(\Phi)=\operatorname{spec}(G)=\\{-\rho(G)^{(1)},0^{(n-2)},\rho(G)^{(1)}\\}$.
Thus $\Phi=(G,\varphi)$ is a bipartite $\mathbb{T}$-gain graph with exactly
one positive eigenvalue. Therefore, by Theorem 2.8, $\Phi$ is balanced
complete bipartite.

If $-\Phi$ is balanced, then $-\Phi$ is balanced complete bipartite. By
Theorem 2.7, we have $-\Phi$ is balanced implies $\Phi$ is balanced. Thus
$\Phi$ is balanced complete bipartite.

Conversely, if $\Phi$ is balanced complete bipartite $\mathbb{T}$-gain graph,
then
$\operatorname{spec}(\Phi)=\\{-\rho(G)^{(1)}\break,0^{(n-2)},\rho(G)^{(1)}\\}$.
Hence, equality occurs in (4). ∎

Next, we give a lower bound for the energy in terms of the number of edges and
the spectral radius of the underlying graph.

######  Theorem 3.2.

Let $\Phi=(G,\varphi)$ be a connected $\mathbb{T}$-gain graph. Then
$\mathcal{E}(\Phi)\geq\frac{2m}{\rho(G)}$, where $m$ is the number of edges in
$G$. Equality holds if and only if $\Phi$ is balanced complete bipartite.

###### Proof.

Let $\\{v_{1},v_{2},\dots,v_{n}\\}$ be the vertex set of $\Phi$. Then
$\mathcal{E}(\Phi)=\sum\limits_{j=1}^{n}\mathcal{E}_{\Phi}(v_{j})$. By Theorem
3.1,
$\mathcal{E}(\Phi)=\sum\limits_{j=1}^{n}\mathcal{E}_{\Phi}(v_{j})\geq\frac{2m}{\rho(G)}$,
and the equality holds if and only if $\Phi$ is balanced complete bipartite. ∎

The following is the main result of this section.

######  Theorem 3.3.

Let $\Phi=(G,\varphi)$ be a connected triangle-free $\mathbb{T}$-gain graph.
Then$\mathcal{E}(\Phi)\geq 2\delta$, where $\delta$ is the minimum vertex
degree of $G$. Equality holds if and only if $\Phi$ is a balanced complete
bipartite $\mathbb{T}$-gain graph and both the partite sets have size
$\delta$.

###### Proof.

By Theorem 3.1, Theorem 2.1, and Theorem 2.2, we have

$\mathcal{E}(\Phi)=\sum\limits_{j=1}^{n}\mathcal{E}_{\Phi}(v_{j})\geq\sum\limits_{j=1}^{n}\frac{d(v_{j})}{\rho(G)}\geq\frac{n\delta}{\rho(G)}\geq\frac{n\delta}{\sqrt{m}}\geq\frac{n\delta}{\sqrt{\lfloor\frac{n^{2}}{4}\rfloor}}\geq\frac{n\delta}{\sqrt{\frac{n^{2}}{4}}}=2\delta.$
(5)

Equality holds if only if all the terms in the equation (5) are equal. By
Theorem
3.1,$\sum\limits_{j=1}^{n}\mathcal{E}_{\Phi}(v_{j})=\sum\limits_{j=1}^{n}\frac{d(v_{j})}{\rho(G)}$
if and only if $\Phi$ is balanced complete bipartite $\mathbb{T}$-gain graph.
It is easy to see that
$\sum\limits_{j=1}^{n}\frac{d(v_{j})}{\rho(G)}=\frac{n\delta}{\rho(G)}$ if and
only if $G$ is regular with degree $\delta$. By Theorem
2.3(i),$\rho(G)=\sqrt{m}$ if and only if $G$ is a complete bipartite graph. By
Mantel’s
theorem,$\frac{n\delta}{\sqrt{m}}=\frac{n\delta}{\sqrt{\lfloor\frac{n^{2}}{4}\rfloor}}$
if and only if $G$ is bipartite with equal partition size. Combining these
observations, we get $\Phi\sim(K_{\delta,\delta},1)$. Thus
$\mathcal{E}(\Phi)=2\delta$ implies $\Phi\sim(K_{\delta,\delta},1)$. The
converse is easy to verify. ∎

######  Remark 3.1.

Theorem 3.3 holds for signed and mixed graphs as well.

##  4  Upper bound for the energy of gain graphs in terms of the matching
number

In this section, we establish an upper bound for the energy of
$\mathbb{T}$-gain graphs in terms of the matching number of the underlying
graph. This result is the counterpart of the corresponding known result for
the undirected graphs [2, 22, 26]. We characterize all the $\mathbb{T}$-gain
graphs for which the upper bound is attained. We begin this section with
needed results, including the Coulson integral formula for the energy of
$\mathbb{T}$-gain graphs and a relationship between the characteristic and the
matching polynomials of $\mathbb{T}$-gain graphs.

The $\mathbb{T}$-gain energy of $\mathbb{T}$-gain graph $\Phi$ is the sum of
the singular values of $A(\Phi)$. This section provides an integral
representation for the $\mathbb{T}$-gain energy. Using this representation, we
establish a bound for the $\mathbb{T}$-gain energy. Let $\Phi=(G,\varphi)$ be
a $\mathbb{T}$-gain graph, and let $P_{\Phi}(x):=\det(xI-A(\Phi))$ be the
characteristic polynomial of $\Phi$.

Let $\Phi$ be a $\mathbb{T}$-gain graph on $n$ vertices with the
characteristic polynomial $P_{\Phi}(x)$. Then, by Theorem 2.9,

$\displaystyle\mathcal{E}(\Phi)=\frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{1}{t^{2}}\log\bigg{|}t^{n}P_{\Phi}\left(\frac{i}{t}\right)\bigg{|}dt.$
(6)

Let
$P_{\Phi}(x)=x^{n}+b_{1}(\Phi)x^{n-1}+b_{2}(\Phi)x^{n-2}+\cdots+b_{n}(\Phi)$
be the characteristic polynomial of $\Phi$. Then simplifying the expression
(6), we have

$\displaystyle\mathcal{E}(\Phi)$
$\displaystyle=\frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{1}{t^{2}}\log\bigg{|}1+t^{n}\sum\limits_{k=1}^{n}b_{k}(\Phi)\left(\frac{i}{t}\right)^{n-k}\bigg{|}dt$
$\displaystyle=\frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{1}{t^{2}}\log\bigg{|}1+t^{n}\sum\limits_{k=1}^{n}(-i)^{k}b_{k}(\Phi)t^{k-n}\bigg{|}dt$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\frac{1}{t^{2}}\log\bigg{\\{}A(t)^{2}+B(t)^{2}\bigg{\\}}dt,$
(7)

where
$A(t)=\bigg{(}1+\sum\limits_{k=1}^{\lfloor\frac{n}{2}\rfloor}(-1)^{k}b_{2k}(\Phi)t^{2k}\bigg{)}$
and
$B(t)=\bigg{(}\sum\limits_{k=0}^{\lfloor\frac{n-1}{2}\rfloor}(-1)^{k}b_{2k+1}(\Phi)t^{2k+1}\bigg{)}$.

The above discussion gives a proof for the following theorem.

######  Theorem 4.1.

Let $\Phi$ be a $\mathbb{T}$-gain graph with characteristic polynomial
$P_{\Phi}(x)=x^{n}+b_{1}(\Phi)x^{n-1}+b_{2}(\Phi)x^{n-2}+\cdots+b_{n}(\Phi).$
Then

$\mathcal{E}(\Phi)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\frac{1}{t^{2}}\log\Bigg{\\{}\bigg{(}1+\sum\limits_{k=1}^{\lfloor\frac{n}{2}\rfloor}(-1)^{k}b_{2k}(\Phi)t^{2k}\bigg{)}^{2}+\bigg{(}\sum\limits_{k=0}^{\lfloor\frac{n-1}{2}\rfloor}(-1)^{k}b_{2k+1}(\Phi)t^{2k+1}\bigg{)}^{2}\Bigg{\\}}dt.$
(8)

For a graph $G$, let $m(G,j)$ be the number of matchings of size $j$ in $G$.
The matching polynomial of a graph $G$ on $n$ vertices, denoted by $m_{G}(x)$,
is defined as follows

$m_{G}(x)=\sum\limits_{j=0}^{\lfloor\frac{n}{2}\rfloor}(-1)^{j}m(G,j)x^{n-2j}.$

Define $m(G,0)=1$. A matching in a $\mathbb{T}$-gain graph $\Phi=(G,\varphi)$
is the matching in the underlying graph $G$, and the matching polynomial of
$\Phi$, denoted by $m_{\Phi}(x)$, is defined as $m_{\Phi}(x)=m_{G}(x)$.

For a subgraph $K$ of a graph $G$, let $n(K)$ denote the number of components
in $K$, and $G-K$ is the graph obtained from $G$ by deleting the vertices of
$K$. In [9], Godsil and Gutman established the following relationship between
the characteristic polynomial and the matching polynomial of a simple graph
$G$.

######  Theorem 4.2 ([9, Theorem 4]).

Let $P_{G}(x)$ and $m_{G}(x)$ be the characteristic and the matching
polynomials of a simple graph $G$, respectively. Then

$P_{G}(x)=m_{G}(x)+\sum\limits_{K}(-2)^{n(K)}m_{G-K}(x),$

where the summation is over all nontrivial subgraphs $K$ of $G$, which are the
union of vertex disjoint cycles ( i.e., $2$-regular subgraphs of $G$).

Our next objective is to extend the above theorem for the $\mathbb{T}$-gain
graphs.

######  Theorem 4.3.

Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph, and $P_{\Phi}(x)$ and
$m_{G}(x)$ be the characteristic and the matching polynomials of $\Phi$,
respectively. Then

$P_{\Phi}(x)=m_{G}(x)+\sum\limits_{K}(-2)^{n(K)}\prod\limits_{C\in
K}\operatorname{Re}(\varphi(C))m_{G-K}(x),$

where the summation is over all nontrivial subgraphs $K$ of $G$ which are
union of vertex disjoint cycles, and the product is over all cycles $C$ in
$K$.

###### Proof.

Let $P_{\Phi}(x)=x^{n}+\sum\limits_{j=1}^{n}b_{j}(\Phi)x^{n-j}$ be the
characteristic polynomial of $\Phi$. By Lemma 2.3,

$P_{\Phi}(x)=\sum\limits_{j=0}^{n}\left(\sum\limits_{H\in\mathcal{H}_{j}}(-1)^{n(H)}2^{c(H)}\prod\limits_{C\in\mathcal{C}(H)}\operatorname{Re}(\varphi(C))\right)x^{n-j}.$
(9)

Note that $H$ is an arbitrary elementary subgraph of $G$ with $j$ number of
vertices. Then, each component of $H$ is either a cycle or an edge. Split $H$
into two subgraphs $K$ and $F$ such that $K$ contains all the cycles (vertex
disjoint) and $F$ contains the remaining edges (independent edges) of $H$,
respectively. Therefore, $H=F\cup K$ such that the number of vertices $j$ in
$H$ can be expressed as $j=2n(F)+|K|$, where $|K|$ denotes the number of
vertices in $K$. By rearranging the internal summation of (9) in terms of $K$
and $F$, we get

$P_{\Phi}(x)=\sum\limits_{j=0}^{n}\sum\limits_{K}\sum\limits_{F}(-2)^{n(K)}(-1)^{n(F)}\prod\limits_{C\in
K}\operatorname{Re}(\varphi(C))x^{n-(2n(F))+|K|)},$

where $j=2n(F)+|K|$. Now, consider $K$ as an arbitrary subgraph of $G$ such
that it contains only vertex-disjoint cycles. After reordering the summations,
we get

$\displaystyle P_{\Phi}(x)$
$\displaystyle=\sum\limits_{K}(-2)^{n(K)}\prod\limits_{C\in
K}\operatorname{Re}(\varphi(C))\sum\limits_{n(F)=0}^{\lfloor\frac{n-|K|}{2}\rfloor}\sum\limits_{F}(-1)^{n(F)}x^{n-(2n(F)+|K|)}$
$\displaystyle=\sum\limits_{K}(-2)^{n(K)}\prod\limits_{C\in
K}\operatorname{Re}(\varphi(C))\sum\limits_{n(F)=0}^{\lfloor\frac{n-|K|}{2}\rfloor}m(G-K,n(F))(-1)^{n(F)}x^{n-(2n(F)+|K|)}$
$\displaystyle=\sum\limits_{K}(-2)^{n(K)}\prod\limits_{C\in
K}\operatorname{Re}(\varphi(C))\sum\limits_{t=0}^{\lfloor\frac{n-|K|}{2}\rfloor}(-1)^{t}m(G-K,t)x^{n-|K|-2t}$
$\displaystyle=\sum\limits_{K}(-2)^{n(K)}\prod\limits_{C\in
K}\operatorname{Re}(\varphi(C))m_{G-K}(x).$

We split the summation into two parts, $K=\phi$ and $K\neq\phi$. Then

$\displaystyle
P_{\Phi}(x)=m_{G}(x)+\sum\limits_{K\neq\phi}(-2)^{n(K)}\prod\limits_{C\in
K}\operatorname{Re}(\varphi(C))m_{G-K}(x)$

where the summation is over all nontrivial subgraph $K$ of $G$ which are union
of vertex disjoint cycles, $n(K)$ is the number of cycles in $K$, and $G-K$ is
the graph obtained from $G$ by deleting the vertices of $K$. ∎

######  Remark 4.1.

The matching polynomial of a simple graph is the same as its characteristic
polynomial if and only if the graph is a forest [9]. By Theorem 4.3, it is
easy to see that if $\Phi=(G,\varphi)$ is a $\mathbb{T}$-gain graph and $G$ is
a forest, then the matching polynomial and the characteristic polynomial are
the same for $\Phi$. However, in general, the converse need not be true. Let
$\Phi=(C_{3},\varphi)$ be a $\mathbb{T}$-gain graph on a triangle
$C_{3}:v_{1}\sim v_{2}\sim v_{3}\sim v_{1}$ such that
$\varphi(\vec{e}_{1,2})=\varphi(\vec{e}_{2,3})=\varphi(\vec{e}_{3,1})=i$. Then
$P_{\Phi}(x)=m_{\Phi}(x)=x^{3}-3x$.

The following result provides a sufficient condition for a $\mathbb{T}$-gain
graph with the same characteristic and matching polynomials.

######  Corollary 4.1.

Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph with
$\operatorname{Re}(\varphi(C))=0$ for all cycle $C$ in $G$. Then

$P_{\Phi}(x)=m_{G}(x)=\sum\limits_{j=0}^{\lfloor\frac{n}{2}\rfloor}(-1)^{j}m(G,j)x^{n-2j}.$

As a consequence of the above result, we have a formula for energy.

######  Corollary 4.2.

Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph with
$\operatorname{Re}(\varphi(C))=0$ for all cycle $C$ in $G$. Then

$\mathcal{E}(\Phi)=\frac{2}{\pi}\int\limits_{0}^{\infty}\frac{1}{x^{2}}\log\bigg{(}1+\sum\limits_{j=1}^{\lfloor\frac{n}{2}\rfloor}m(G,j)x^{2j}\bigg{)}dx.$

###### Proof.

Let
$P_{\Phi}(x)=x^{n}+b_{1}(\Phi)x^{n-1}+b_{2}(\Phi)x^{n-2}+\cdots+b_{n}(\Phi)$
be the characteristic polynomial of $\Phi$. From Corollary 4.1, we have
$P_{\Phi}(x)=\sum\limits_{j=0}^{\lfloor\frac{n}{2}\rfloor}(-1)^{j}m(G,j)x^{n-2j}$.
Comparing coefficients of both the above polynomials, we get
$b_{2j}(\Phi)=(-1)^{j}m(G,j)$ and $b_{2j+1}(\Phi)=0$, for
$j=0,1,\dots,\lfloor\frac{n}{2}\rfloor$. Now, the result follows from Theorem
4.1. ∎

Corollary 4.2 is useful in constructing equienergetic $\mathbb{T}$-gain graphs
which are not switching equivalent. Two $\mathbb{T}$-gain graphs $\Phi_{1}$
and $\Phi_{2}$ are  equienergetic if
$\mathcal{E}(\Phi_{1})=\mathcal{E}(\Phi_{2})$. Let $G$ be a graph with cycles
that are vertex disjoint. If $\Phi_{1}=(G,\varphi_{1})$ and
$\Phi_{2}=(G,\varphi_{2})$ are two $\mathbb{T}$-gain graphs such that
$\operatorname{Re}(\varphi_{j}(C))=0$, $j=1,2$ for all cycles $C$ in $G$, then
$\Phi_{1}$ and $\Phi_{2}$ are equienergetic.

Define the graphs $G_{p,q,r}$ and $T_{p,q}$, with $p,q,r\geq 0$ as follows:
$G_{p,q,r}$ is a graph on $(p+q+r+2)$ vertices with $r$ triangles with a
common base edge, say $e=\\{u,v\\}$, and the vertex $u$ is adjacent to $p$
pendent vertices, and $v$ is adjacent to $q$ pendent vertices. The tree
$T_{p,q}$ consists of $(p+q+2)$ vertices, where there is an edge $e=\\{u,v\\}$
such that $u$ is adjacent to $p$ pendent vertices, and $v$ is adjacent to $q$
pendent vertices. Both the graphs are depicted in Figure 2.

Figure 2:  Graphs $G_{p,q,r}$ and $T_{p,q}$

Our next objective is to obtain bounds for the energy of a $\mathbb{T}$-gain
graph in terms of the matching number. The corresponding result is known for
undirected graphs [2, Theorem 12]. The following key result is used in the
proof of [2, Theorem 12].

######  Lemma 4.1 ([22]).

For any positive integer $r$,
$\mathcal{E}(G_{p,q,r})<\mathcal{E}(T_{p+r,q+r}).$

Unfortunately, the above result is not true for $\mathbb{T}$-gain graphs.

######  Example 4.1.

Consider two graphs $\Phi=(G_{1,1,2},\varphi)$ and $T_{3,3}$ given in Figure
3. For the $\mathbb{T}$-gain graph $\Phi$,
$\varphi(\vec{e}_{1,2})=\varphi(\vec{e}_{2,1})=-1$, and gain of all other
edges are $1$. Then $\mathcal{E}(\Phi)\not<\mathcal{E}(T_{3,3})$.

Figure 3:  $\mathbb{T}$-gain graph $\Phi=(G_{1,1,2},\varphi)$ and $T_{3,3}$

Nevertheless, we have the following result for $\mathbb{T}$-gain graphs, which
is good enough for our purpose.

######  Lemma 4.2.

Let $\Phi=(G_{p,q,1},\varphi)$ be any $\mathbb{T}$-gain graph. Then
$\mathcal{E}(\Phi)<\mathcal{E}(T_{p+1,q+1}).$

###### Proof.

Let $\Phi=(G_{p,q,1},1)$ be a $\mathbb{T}$-gain graph on the underlying graph
$G_{p,q,1}$ given in Figure 2. Then $\Phi$ contains exactly one cycle of
length $3$, say $C$. Let $k=p+q+2$. Then $k\geq 2$. Define $\Phi_{1}=\Phi\cup
K_{1}$. Then $\mathcal{E}(\Phi)=\mathcal{E}(\Phi_{1})$. Here
$|V(\Phi_{1})|=|V(T_{p+1,q+1})|=k+2$. Since $T_{p+1,q+1}$ is a tree, so

$P_{T_{p+1,q+1}}(x)=m_{T_{p+1,q+1}}(x)=x^{k+2}-(k+1)x^{k}+(1+p+q+pq)x^{k-2}.$

Also, by Theorem 4.3, we have

$\displaystyle P_{\Phi_{1}}(x)$
$\displaystyle=m_{\Phi_{1}}(x)+(-2)^{1}\operatorname{Re}(\varphi(C))m_{\Phi_{1}-C}(x)$
$\displaystyle=x^{k+2}-(k+1)x^{k}+(pq+p+q)x^{k-2}-2m_{(k-1)K_{1}}(x)\operatorname{Re}(\varphi(C))$
$\displaystyle=x^{k+2}-(k+1)x^{k}+(pq+p+q)x^{k-2}-2x^{k-1}\operatorname{Re}(\varphi(C)).$

By Coulson integral formula (8) for the $\mathbb{T}$-gain graphs, we have

$\mathcal{E}(T_{p+1,q+1})=\frac{1}{2\pi}\int\limits_{-\infty}^{+\infty}\frac{1}{x^{2}}\log\left(f_{1}(x)\right)dx$
$\mathcal{E}(\Phi_{1})=\frac{1}{2\pi}\int\limits_{-\infty}^{+\infty}\frac{1}{x^{2}}\log\left(f_{2}(x)\right)dx,$

where

$f_{1}(x)=\left(1+(k+1)x^{2}+(1+p+q+pq)x^{4}\right)^{2}$

and

$f_{2}(x)=\left(1+(k+1)x^{2}+(pq+p+q)x^{4}\right)^{2}+\left(2\operatorname{Re}(\varphi(C))x^{3}\right)^{2}.$

Then,

$\displaystyle f_{1}(x)-f_{2}(x)$
$\displaystyle=x^{4}\\{2+2(k+1)x^{2}+2(1+p+q+pq)x^{4}-x^{4}\\}-4x^{6}\operatorname{Re}(\varphi(C))^{2}.$

Since $|\operatorname{Re}(\varphi(C))|\leq 1$, so

$\displaystyle f_{1}(x)-f_{2}(x)$ $\displaystyle\geq
x^{4}\\{2+2(k+1)x^{2}+(1+2p+2q+2pq)x^{4}\\}-4x^{6}$
$\displaystyle=x^{4}\\{2+(2p+2q+2)x^{2}+(1+2p+2q+2pq)x^{4}\\}>0.$

Now $f_{1}(x)-f_{2}(x)=0$ if and only if $x=0$. Therefore, for any non-empty
graph, $\mathcal{E}(\Phi)<\mathcal{E}(T_{p+1,q+1}).$ ∎

Let us consider the tree $T_{1}$ of diameter $3$ and the maximum edge degree
$\Delta_{e}$ given in Figure 1 (see Section 2). Now, we compare the energies
of $(G_{p,q,1},\varphi)$ and $T_{1}$, where $p,q\geq 0$.

######  Lemma 4.3.

Let $\Phi=(G_{p,q,1},\varphi)$ be a $\mathbb{T}$-gain graph, and $T_{1}$ be a
tree given in Figure 1 such that the maximum edge degree of $\Phi$ is less
than or equal to the maximum edge degree of $T_{1}$. Then

$\mathcal{E}(\Phi)<\mathcal{E}(T_{1}).$

###### Proof.

Let $\Phi=(G_{p,q,1},\varphi)$ be a $\mathbb{T}$-gain graph on $G_{p,q,1}$. By
Lemma 4.2, we have $\mathcal{E}(\Phi)<\mathcal{E}(T_{p+1,q+1})$. Note that, by
Lemma 2.1,
$\mathcal{E}(T_{p+1,q+1})\leq\mathcal{E}(T_{\lfloor\frac{p+q+2}{2}\rfloor,\lceil\frac{p+q+2}{2}\rceil})$.
Consider the tree $T_{1}$ given in Figure 1, where $\Delta_{e}(T_{1})$ is the
maximum edge degree of $T_{1}$. Since$\Delta_{e}(T_{1})\geq p+q+2$, so
$T_{\lfloor\frac{p+q+2}{2}\rfloor,\lceil\frac{p+q+2}{2}\rceil}$ is an induced
subgraph of $T_{1}$. Therefore, by Theorem 2.10,
$\mathcal{E}(T_{\lfloor\frac{p+q+2}{2}\rfloor,\lceil\frac{p+q+2}{2}\rceil})\leq\mathcal{E}(T_{1})$.
By combining the above three inequalities, we have
$\mathcal{E}(\Phi)<\mathcal{E}(T_{1})$. ∎

Next, we compare the energies of $(T_{p,q},\varphi)$ and $T_{1}$, where
$p,q\geq 0$.

######  Lemma 4.4.

Let $\Phi=(T_{p,q},\varphi)$ be a $\mathbb{T}$-gain graph and $T_{1}$ be a
tree given in Figure 1 such that the maximum edge degree of $\Phi$ is less
than or equal to the maximum edge degree of $T_{1}$. Then
$\mathcal{E}(\Phi)\leq\mathcal{E}(T_{1})$. Equality holds if and only if
$T_{p,q}$ is isomorphic to $T_{1}$.

###### Proof.

By Lemma 2.1,
$\mathcal{E}(\Phi)=\mathcal{E}(T_{p,q})\leq\mathcal{E}(T_{\lfloor\frac{p+q}{2}\rfloor,\lceil\frac{p+q}{2}\rceil})$.
Let $\Delta_{e}$ be the maximum edge degree of $T_{1}$. Since
$p+q\leq\Delta_{e}$, so
$T_{\lfloor\frac{p+q}{2}\rfloor,\lceil\frac{p+q}{2}\rceil}$ is an induced
subgraph of $T_{1}$. Therefore, by Theorem 2.10,
$\mathcal{E}(T_{\lfloor\frac{p+q}{2}\rfloor,\lceil\frac{p+q}{2}\rceil})\leq\mathcal{E}(T_{1})$.
Thus, $\mathcal{E}(\Phi)\leq\mathcal{E}(T_{1})$. If
$\mathcal{E}(\Phi)=\mathcal{E}(T_{1})$, then all the intermediate inequalities
will be equal. Therefore, by Theorem 2.10 and Lemma 2.1, $T_{p,q}$ is
isomorphic to $T_{1}$. ∎

In [2], Akbari et al. decomposed an undirected graph $G$, with matching number
$\mu$, into $\mu$ copies edge-disjoint copies of $G_{p,q,r}$, where $r\geq 0$.
In the next lemma, we decompose $G$ into $\mu$ edge-disjoint copies of either
$G_{p,q,0}$ or $G_{p,q,1}$. This result is crucial in proving the main result
of this section.

######  Lemma 4.5.

Any graph $G$ with matching number $\mu$ can be decomposed into $\mu$ edge-
disjoint copies either $T_{p,q}$ or $G_{p,q,1}$.

###### Proof.

Let $G$ be a graph with matching number $\mu$. Let
$\mathcal{M}=\\{e_{1},e_{2},\dots,e_{\mu}\\}$ be a maximum matching of $G$.
Let $H_{j}$ be a subgraph of $G$ containing $e_{j}$ and its all incidence
edges, for $j=1,2,\cdots,\mu$. Take $S_{1}=H_{1}$ and
$S_{j}=H_{j}\setminus\bigcup\limits_{i=1}^{j-1}E(H_{i})$, for
$j=2,3,\dots,\mu$. Note that each $S_{j}$ is of the form $G_{p,q,r}$, where
$r\geq 0$.

For $j=1,2,\dots,\mu$, construct the subgraph $L_{j}$ from the graph $S_{j}$
as follows: Suppose two of the nonmatching edges of any triangle in $S_{j}$
are incident with a matching edge $e_{i}$, for $i\neq j$, then remove the
edges from $S_{j}$ and add them to $S_{i}$. That is, if the union $S_{j}\cup
S_{i}$ contains the subgraph $K$ ( see Figure 4), then keep the edges
$e_{x,t}$ and $e_{y,t}$ in the subgraph $S_{i}$, and discard these edges in
$S_{j}$. Continue this process until none of the modified subgraphs contain a
triangle with two of its edges incident with two matching edges in the
subgraph. For $j=1,2\dots,\mu$, let $L_{j}$ denote the subgraph containing the
matching edge $e_{j}$ obtained at the end of the above process. It is easy to
see that all subgraphs $L_{j}$ are edge-disjoint, and their union is $G$.

Figure 4:

Next, we claim that, each $L_{j}$ is of the form either $T_{p,q}$ or
$G_{p,q,1}$. For, suppose that $L_{j}$ contains more than one triangle. Let
$v_{1},v_{2},\dots,v_{r}$ be the vertices of the triangles (see graph $M$ in
Figure 4), where $r\geq 2$. From the construction of $L_{j}$, none of the
vertices $v_{1},v_{2},\dots,v_{r}$ is adjacent to any matching edges in
$\mathcal{M}$. Therefore, the collection
$\mathcal{M^{\prime}}=\\{e_{1},e_{2},\dots,e_{j-1},e_{j+1},\dots,e_{\mu},e_{x,1},e_{y,2}\\}$
is a matching, and it contains more edges than $\mathcal{M}$, which is a
contradiction. Thus, the result follows. ∎

Now, we are ready to prove the main result of this section, which gives a
bound for the energy in terms of matching number and maximum edge degree. Let
$P_{n}$ denote the path on $n$ vertices.

######  Theorem 4.4.

Let $\Phi=(G,\varphi)$ be any $\mathbb{T}$-gain graph on a simple graph $G$
with the matching number $\mu$ and the maximum edge degree $\Delta_{e}$. Then
we have the following:

  * (A)

If $\Delta_{e}$ is even, then $\mathcal{E}(\Phi)\leq
2\mu\sqrt{2\Delta_{e}+1}$. Equality holds if and only if $\Phi$ is the
disjoint union of $\mu$ copies of $(P_{2},\varphi)$ and some isolated
vertices.

  * (B)

If $\Delta_{e}$ is odd, then
$\mathcal{E}(\Phi)\leq\mu\left(\sqrt{b+2\sqrt{b}}+\sqrt{b-2\sqrt{b}}\right)$,
where $b=2(\Delta_{e}+1)$. Equality holds if and only if $\Phi$ is $\mu$
copies of $(P_{3},\varphi)$ and some isolated vertices.

###### Proof.

Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph with matching number
$\mu$. Let $\mathcal{M}=\\{e_{1},e_{2},\cdots,e_{\mu}\\}$ be a maximum
matching of $\Phi$, and let $d(e_{j})$ denote the edge degree of $e_{j}$ in
$\Phi$ for $j=1,2,\cdots,\mu$. Then, by Lemma 4.5, we can decompose $\Phi$
into edge-disjoint subgraphs $(L_{j},\varphi)$, for $j=1,2,\dots,\mu$. As
$A(\Phi)=\sum\limits_{j=1}^{\mu}A(L_{j},\varphi)$, by Theorem 2.11, we get
$\mathcal{E}(\Phi)\leq\sum\limits_{j=1}^{\mu}\mathcal{E}(L_{j},\varphi)$. Now
each subgraph $L_{j}$ is of the form $G_{p_{j},q_{j},r}$, with either $r=0$ or
$r=1$, and $p_{j}+q_{j}+2r\leq d(e_{j})\leq\Delta_{e}$. Now, by Lemma 4.3 and
Lemma 4.4, $\mathcal{E}(G_{p_{j},q_{j},r},\varphi)\leq\mathcal{E}(T_{1})$, for
$j=1,2,\cdots,\mu$, where $T_{1}$ is given in Figure 1. Thus
$\mathcal{E}(\Phi)\leq\mu\mathcal{E}(T_{1})$. Hence, by Lemma 2.2, the
inequalities in $(A)$ and $(B)$ follows.

Suppose $\mathcal{E}(\Phi)=\mu\mathcal{E}(T_{1})$. Then
$\mathcal{E}(L_{j},\varphi)=\mathcal{E}(T_{1})$, for $j=1,2,\cdots,\mu$. Also,
by Lemma 4.3, $L_{j}$ can not be the form of $G_{p_{j},q_{j},1}$. Thus each
$L_{j}$ is of the from $T_{p_{j},q_{j}}$ for $j=1,2,\cdots,\mu$. By Lemma 4.4,
$L_{j}=T_{1}$, for $j=1,2,\dots,\mu$. Each subgraph $L_{j}$ is a component of
$G$. For, suppose that $L_{j}$ is not a connected component of $G$. Then there
exists an edge $\bar{e}$ incident with both the matching edges $e_{j}$ and
$e_{t}$ for some $t\neq j$, otherwise we can find a matching larger than
$\mu$. Without loss of generality, let us assume that $\bar{e}$ is an edge in
$L_{t}$. The degree of $e_{j}$ in $L_{j}$ is less than or equal to
$d(e_{j})-1$. Since $L_{j}=T_{1}$, so the degree of $e_{j}$ in $L_{j}$ is
$\Delta_{e}$. Therefore, $\Delta_{e}\leq d(e_{j})-1\leq\Delta_{e}-1$, which is
a contradiction. So $L_{j}$ is a connected component. Therefore all the
subgraphs $L_{j}$, for $j=1,2,\dots,\mu$ are connected components of $G$ with
maximum edge degree $\Delta_{e}$. Suppose that $\Delta_{e}\geq 2$ for some
matching edge $e$. Then, by removing the edge $e$ from $\mathcal{M}$ and
adding the two edges that are incident with the end vertices of $e$, we get a
matching size larger than $\mu$; this is not possible, as $\mathcal{M}$ is a
maximum matching. Thus $\Delta_{e}\leq 1$. If $\Delta_{e}=1$, then
$T_{1}=P_{3}$ and if $\Delta_{e}=0$, then $T_{1}=P_{2}$. Hence, the result
follows. ∎

######  Corollary 4.3.

Let $\Phi=(G,\varphi)$ be any $\mathbb{T}$-gain graph on a simple graph $G$
with the matching number $\mu$ and the maximum edge degree $\Delta_{e}$. Then

$\mathcal{E}(\Phi)\leq 2\mu\sqrt{2\Delta_{e}+1}.$

###### Proof.

Note that, if $c=\sqrt{b+2\sqrt{b}}+\sqrt{b-2\sqrt{b}}>0$, where
$b=2(\Delta_{e}+1)$, then $c^{2}=2b+2\sqrt{(b-2)^{2}-4}<4(b-1)$. Therefore,
$\sqrt{b+2\sqrt{b}}+\sqrt{b-2\sqrt{b}}<2\sqrt{2\Delta_{e}+1}$. ∎

######  Remark 4.2.

Hermitian adjacency matrix of a mixed graph and adjacency matrix of a signed
graph are particular cases of the adjacency matrix of a $\mathbb{T}$-gain
graph. Therefore, all the above results also hold for mixed and signed graphs.

##  5 Extremal families for the energy of unicyclic $\mathbb{T}$-gain graphs

For any simple graph $G$, let $\mathcal{T}_{G}$ denote the collection of all
$\mathbb{T}$-gain graphs on the underlying graph $G$. This section studies the
extremum problem for the energies of $\mathbb{T}$-gain graphs in
$\mathcal{T}_{G}$. In [12], Haemers conjectured that among all the signed
graphs defined of $K_{n}$, the complete graph on $n$ vertices, all one edge
weights has the least energy. That is,
$\mathcal{E}(K_{n})\leq\mathcal{E}(K_{n},\psi)$ for any signed graph
$(K_{n},\psi)$. In [1], Akbari et al. proved this conjecture. In this section,
first, we observe that the counterpart of this result for $\mathbb{T}$-gain
graphs need not be true.

######  Example 5.1.

Let $\Phi=(K_{3},\varphi)$ be a $\mathbb{T}$-gain graph on $K_{3}$ with vertex
set $\\{v_{1},v_{2},v_{3}\\}$ such that $\varphi(\vec{e}_{1,2})=i$,
$\varphi(\vec{e}_{2,3})=\varphi(\vec{e}_{3,1})=1$. Then
$A(\Phi)=\begin{pmatrix}0&i&1\\\ -i&0&1\\\ 1&1&0\end{pmatrix}$. Thus
$\mathcal{E}(\Phi)=2\sqrt{3}\approx 3.464$. Also $\mathcal{E}(K_{3})=4$. So
$\mathcal{E}(K_{3})>\mathcal{E}(K_{3},\varphi)$.

From the following example one can see that for a graph $G$ the energy
$\mathcal{E}(G)$ need not be the minimum/maximum energy among all
$\mathbb{T}$-gain graphs defined on $G$.

######  Example 5.2.

In Figure 5, we can see that $\Phi_{1}$ and $\Phi_{2}$ are two
$\mathbb{T}$-gain graphs on a same underlying graph $G$, where
$\mathcal{E}(\Phi_{1})=6.133,\mathcal{E}(\Phi_{2})=5.9939$ and
$\mathcal{E}(G)=6.0409$. Thus,
$\mathcal{E}(\Phi_{2})<\mathcal{E}(G)<\mathcal{E}(\Phi_{1})$.

Figure 5:  The gain graphs $\Phi_{1}$ and $\Phi_{2}$ have the same underlying
graph $G$.

Motivated by this observation, it is natural to study the graphs for which the
energy of the underlying graph attains either maximum or minimum energy among
the energies of all $\mathbb{T}$-gain graphs defined on $G$. In [24], we
proved that $\mathcal{E}(K_{n,n})\leq\mathcal{E}(K_{n,n},\varphi)$, where
$(K_{n,n},\varphi)$ is any $\mathbb{T}$-gain graph on $K_{n,n}$. Here, we
present another class of graphs, namely unicyclic graphs, for which its energy
attains extremum energy overall $\mathbb{T}$-gain graphs associated with it.
The girth of a graph $G$ is the length of a shortest cycle in $G$.

######  Theorem 5.1.

Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph on a unicyclic graph $G$
with girth $r$.

  1. (i)

If $r\equiv 0(\mod 4)$, then $\mathcal{E}(\Phi)\geq\mathcal{E}(G)$. Further,
equality holds if and only if $\Phi$ is balanced.

  2. (ii)

If $r\equiv 1(\mod 4)$ or $r\equiv 3(\mod 4)$, then
$\mathcal{E}(\Phi)\leq\mathcal{E}(G)$. Further, equality holds if and only if
either $\Phi$ or $-\Phi$ is balanced.

  3. (iii)

If $r\equiv 2(\mod 4)$, then $\mathcal{E}(\Phi)\leq\mathcal{E}(G)$. Further,
equality holds if and only if $\Phi$ is balanced.

###### Proof.

Let $C_{r}$ be the unique cycle in $G$. If
$P_{\Phi}(x)=x^{n}+b_{1}(\Phi)x^{n-1}+b_{2}(\Phi)x^{n-2}+\cdots+b_{n}(\Phi)$
is the characteristic polynomial of $\Phi$, then, by Lemma 2.3, the
coefficients are given by:

$b_{j}(\Phi)=\sum\limits_{H\in\mathcal{H}_{j}}(-1)^{n(H)}2^{c(H)}\prod\limits_{C\in
C(H)}\operatorname{Re}(\varphi(C)),~{}~{}~{}j=1,2,\cdots,n.$

Now, we consider the following cases.

Case (i): Let $r\equiv 0$ (mod $4$). Since $r$ is even, so $G$ is bipartite.
Therefore, for any $\Phi=(G,\varphi)$, by Theorem 2.6, the eigenvalues of
$\Phi$ are symmetric about the origin. Then $b_{2k+1}(\Phi)=b_{2k+1}(G)=0$ for
$k=0,1,\cdots,\lfloor\frac{n-1}{2}\rfloor$. Also $(-1)^{k}b_{2k}(\Phi)\geq 0$
and $(-1)^{k}b_{2k}(G)\geq 0$ for $k=1,2,\cdots,\lfloor\frac{n}{2}\rfloor$.
Let $r=4s$ for some positive integer $s$. Then,

$b_{2k}(\Phi)=\left\\{\begin{array}[]{ccc}(-1)^{k}m(\Phi,k)&\mbox{if}&2k<r\\\
(-1)^{k}m(\Phi,k)+(-1)^{k-2s+1}2\operatorname{Re}(\varphi(C_{r}))m(\Phi-
C_{r},k-2s)&\mbox{if}&2k\geq r,\end{array}\right.$

for $k=1,2,\cdots,\lfloor\frac{n}{2}\rfloor.$ Thus
$(-1)^{k}b_{2k}(\Phi)=(-1)^{k}b_{2k}(G)=m(\Phi,k)\geq 0$ for $2k<r$. Also,
$(-1)^{k}b_{2k}(\Phi)=m(\Phi,k)-2\operatorname{Re}(\varphi(C_{r}))m(\Phi-
C_{r},k-2s)$ and $(-1)^{k}b_{2k}(G)=m(\Phi,k)-2m(\Phi-C_{r},k-2s)$, for
$2k\geq r$. Now $\operatorname{Re}(\varphi(C_{r}))\leq 1$, and equality holds
if and only if $\Phi$ is balanced. Then
$(-1)^{k}b_{2k}(\Phi)\geq(-1)^{k}b_{2k}(G)$, for $2k\geq r$ and equality holds
if and only if $\Phi$ is balanced. Thus, by Lemma 4.1,
$\mathcal{E}(\Phi)\geq\mathcal{E}(G)$ and equality holds if and only if $\Phi$
is balanced.

Case (ii): Either $r\equiv 1$ (mod $4$) or $r\equiv 3$ (mod $4$).

Subcase (a): Let $r\equiv 1$ (mod $4$). Then $r=4s+1$ for some nonnegative
integer $s$. Note that $b_{2k}(\Phi)=(-1)^{k}m(\Phi,k)=b_{2k}(G)$ for
$k=1,2,\cdots,\lfloor\frac{n}{2}\rfloor$. Now, for
$k=0,1,\cdots,\lfloor\frac{n-1}{2}\rfloor$,

$b_{2k+1}(\Phi)=\left\\{\begin{array}[]{ccc}0&\mbox{if}&2k+1<r\\\
(-1)^{k-2s+1}2\operatorname{Re}(\varphi(C_{r}))m(\Phi-
C_{r},k-2s)&\mbox{if}&2k+1\geq r\end{array}\right.$

That is,

$(-1)^{k}b_{2k+1}(\Phi)=\left\\{\begin{array}[]{ccc}0&\mbox{if}&2k+1<r\\\
-2\operatorname{Re}(\varphi(C_{r}))m(\Phi-C_{r},k-2s)&\mbox{if}&2k+1\geq
r\end{array}\right.$ (10)

Consider $-G$ as a $\mathbb{T}$-gain graph on $G$ with the gain of each edge
is $-1$. Note that, $\mathcal{E}(G)=\mathcal{E}(-G)$. Using (10), we have

$(-1)^{k}b_{2k+1}(-G)=\left\\{\begin{array}[]{ccc}0&\mbox{if}&2k+1<r\\\
2m(\Phi-C_{r},k-2s)&\mbox{if}&2k+1\geq r\end{array}\right.$ (11)

It is easy to see that $(-1)^{k}b_{2k+1}(-G)=-(-1)^{k}b_{2k+1}(G)\geq 0$.
Since $|\operatorname{Re}(\varphi(C_{r}))|\leq 1$, by comparing (10) and (11),
we have $|(-1)^{k}b_{2k+1}(\Phi)|\leq(-1)^{k}b_{2k+1}(-G)$ for
$k=0,1,\cdots,\lfloor\frac{n-1}{2}\rfloor$. Equality occur if and only if
either $\operatorname{Re}(\varphi(C_{r}))=1$ or
$\operatorname{Re}(\varphi(C_{r}))=-1$. That is, equality holds if and only if
either $\Phi$ is balanced or $-\Phi$ is balanced. Therefore, by Lemma 4.1,
$\mathcal{E}(\Phi)\leq\mathcal{E}(G)$ and equality holds if and only if either
$\Phi$ is balanced or $-\Phi$ is balanced.

Subcase (b): Let $r\equiv 3$ (mod $4$). Let $r=4s+3$ for some nonnegative
integer $s$. Then similar to Subcase (b),
$(-1)^{k}b_{2k}(\Phi)=m(\Phi,k)=(-1)^{k}b_{2k}(G)$, for
$k=1,2,\cdots,\lfloor\frac{n}{2}\rfloor$ and
$|(-1)^{k}b_{2k+1}(\Phi)|\leq(-1)^{k}b_{2k+1}(G)$, for
$k=0,1,\cdots,\lfloor\frac{n-1}{2}\rfloor$. Hence, by Lemma 4.1,
$\mathcal{E}(\Phi)\leq\mathcal{E}(G)$. Further equality holds if and only if
either $\Phi$ is balanced or $-\Phi$ is balanced.

Case(iii): Let $r\equiv 2$ (mod $4$). Since $r$ is even,
$b_{2k+1}(\Phi)=b_{2k+1}(G)=0$ for $k=0,1,\cdots,\lfloor\frac{n-1}{2}\rfloor$.
Also $(-1)^{k}b_{2k}(\Phi)\geq 0$ and $(-1)^{k}b_{2k}(G)\geq 0$ for
$k=1,2,\cdots,\lfloor\frac{n}{2}\rfloor$. Let $r=4s+2$ for some nonnegative
integer $s$. Then similar to previous case,
$(-1)^{k}b_{2k}(\Phi)=(-1)^{k}b_{2k}(G)$ for $2k<r$. Also
$(-1)^{k}b_{2k}(\Phi)\leq(-1)^{k}b_{2k}(G)$ for $2k\geq r$ and equality holds
if and only if $\Phi$ is balanced. Therefore, by Lemma 4.1,
$\mathcal{E}(\Phi)\leq\mathcal{E}(G)$, and $\mathcal{E}(\Phi)=\mathcal{E}(G)$
holds if and only if $\Phi$ is balanced.

∎

We present an immediate consequence of the above result.

######  Corollary 5.1.

Let $\Phi=(C_{n},\varphi)$ be any $\mathbb{T}$-gain graph on a cycle $C_{n}$
of $n$ vertices. Then

  * (i)

If $n\equiv 0$ ($\mod 4$), then $\mathcal{E}(\Phi)\geq\mathcal{E}(C_{n})$.
Equality holds if and only if $\Phi$ is balanced.

  * (ii)

If $n\equiv 1$ ($\mod 4$) or $n\equiv 3$ ($\mod 4$), then
$\mathcal{E}(\Phi)\leq\mathcal{E}(C_{n})$. Equality holds if and only if
either $\Phi$ or $-\Phi$ is balanced.

  * (iii)

If $n\equiv 2$ ($\mod 4$), then $\mathcal{E}(\Phi)\leq\mathcal{E}(C_{n})$.
Equality holds if and only if $\Phi$ is balanced.

For a simple graph $G$, the collection $\mathcal{T}_{G}$ is called
equienergetic if the energy of all $\mathbb{T}$-gain graphs in
$\mathcal{T}_{G}$ are the same. If $T$ is a tree, then $\mathcal{T}_{T}$ is
equienergetic. Next, we consider the problem of characterizing graphs $G$ for
which the family $\mathcal{T}_{G}$ is equienergetic.

######  Theorem 5.2.

If $G$ is a graph with all its cycles are vertex disjoint, then
$\mathcal{T}_{G}$ is not equienergetic.

###### Proof.

Let $G$ be a graph with vertex disjoint cycles, and let
$C_{1},C_{2},\dots,C_{p}$ be the only cycles in $G$. Let $\Phi_{1}$ and
$\Phi_{2}$ be two $\mathbb{T}$-gain graphs defined on $G$. Let
$P_{\Phi_{j}}(x)=x^{n}+b_{1}(\Phi_{j})x^{n-1}+b_{2}(\Phi_{j})x^{n-2}+\cdots+b_{n}(\Phi_{j})$
be the characteristic polynomial of $\Phi_{j}$, for $j=1,2$. By Lemma 4.1,

$\mathcal{E}(\Phi_{j})=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\frac{1}{t^{2}}\log\Bigg{\\{}A_{j}(t)^{2}+B_{j}(t)^{2}\Bigg{\\}}dt,$

where
$A_{j}(t)=\bigg{(}1+\sum\limits_{k=1}^{\lfloor\frac{n}{2}\rfloor}(-1)^{k}b_{2k}(\Phi_{j})t^{2k}\bigg{)},B_{j}(t)=\bigg{(}\sum\limits_{k=0}^{\lfloor\frac{n-1}{2}\rfloor}(-1)^{k}b_{2k+1}(\Phi_{j})t^{2k+1}\bigg{)}$
for $j=1,2$, and $b_{k}(\Phi_{j})^{\prime}s$ for $k=1,2,\cdots,n$ are given by
(1). Now, we consider the following three cases. In each of the cases, we find
the choices for the gains $\Phi_{1}$ and $\Phi_{2}$ that serve the purpose:

Case (a): Suppose every cycle in $G$ has odd length. Define $\Phi_{1}$ and
$\Phi_{2}$ as follows:
$\varphi_{1}(\vec{C_{1}})=-\frac{1}{2}+i\frac{\sqrt{3}}{2}$,
$\varphi_{1}(\vec{C_{k}})=i$ for $k=2,3,\dots,p$; and
$\varphi_{2}(\vec{C_{1}})=-1$, $\varphi_{2}(\vec{C_{k}})=i$ for
$k=2,3,\dots,p.$ Then, $A_{1}(t)^{2}=A_{2}(t)^{2}$ and $B_{1}(t)^{2}\neq
B_{2}(t)^{2}$. Thus, $\mathcal{E}(\Phi_{1})\neq\mathcal{E}(\Phi_{2})$.

Case (b): Suppose every cycle in $G$ has even length. Define $\Phi_{1}$ and
$\Phi_{2}$ as follows:
$\varphi_{1}(\vec{C_{1}})=-\frac{1}{2}+i\frac{\sqrt{3}}{2}$,
$\varphi_{1}(\vec{C_{k}})=i$ for $k=2,3,\dots,p$. Also
$\varphi_{2}(\vec{C_{1}})=-1$, $\varphi_{2}(\vec{C_{k}})=i$ for
$k=2,3,\dots,p.$ Then, $B(t)=B^{{}^{\prime}}(t)=0$ and $A_{1}(t)^{2}\neq
A_{2}(t)^{2}$. Thus, $\mathcal{E}(\Phi_{1})\neq\mathcal{E}(\Phi_{2})$.

Case (c): Suppose $G$ contains both the odd and even cycles. Let $C_{1}$ be an
even cycle. Define $\Phi_{1}$ and $\Phi_{2}$ as follows:
$\varphi_{1}(\vec{C})=\varphi_{2}(\vec{C})=i$ for all cycles $C$ except
$C_{1}$ and define
$\varphi_{1}(\vec{C_{1}})=-\frac{1}{2}+i\frac{\sqrt{3}}{2}$,
$\varphi_{2}(\vec{C_{1}})=-1$. The rest of the proof of this case is similar
to that of Case (b). ∎

Given a graph $G$, in [18], we established that the spectrum is the same for
all the elements of $\mathcal{T}_{G}$ if and only if $G$ is a tree. Based on
the above results, we conjecture the following:

######  Conjecture 5.1.

For a connected simple graph $G$, if $\mathcal{T}_{G}$ is equienergetic then
$G$ is a tree.

## Acknowledgments

Aniruddha Samanta thanks the National Board for Higher Mathematics (NBHM),
Department of Atomic Energy, India, for financial support in the form of an
NBHM Post-doctoral Fellowship (Sanction Order No.
0204/21/2023/R$\&$D-II/10038). M. Rajesh Kannan would like to thank the SRC,
IIT Hyderabad for financial support.

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