# Adjacency energy of Hypergraphs

Kauê Cardoso  Instituto Federal do Rio Grande do Sul - Campus Feliz, CEP
95770-000, Feliz, RS, Brazil<EMAIL_ADDRESS>,  Renata Del-
Vecchio  Instituto de Matemática e Estatística, UFF, CEP 24210-200, Niterói,
RJ, Brazil<EMAIL_ADDRESS>,  Lucas Portugal  Graduate program in
Mathematics, UFF, CEP 24210-200, Niterói, RJ, Brazil<EMAIL_ADDRESS>and  Vilmar Trevisan  Instituto de Matemática e Estatística, UFRGS, CEP
91509-900, Porto Alegre, RS, Brazil and
Department of Mathematics and Applications, University of Naples Federico II,
Italy<EMAIL_ADDRESS>
###### Abstract.

In this paper, we define and obtain several properties of the (adjacency)
energy of a hypergraph. In particular, bounds for this energy are obtained as
functions of structural and spectral parameters, such as Zagreb index and
spectral radius. We also study how the energy of a hypergraph varies when a
vertex/edge is removed or when an edge is divied. In addition, we solved the
extremal problem energy for the class of hyperstars, and show that the energy
of a hypergraph is never an odd number.

Keywords. Hypergraphs; Adjacency energy; Edge division; Power graph.

AMS classification. 05C65; 05C50; 15A18.

##  1\. Introduction

In Spectral graph theory, the structure of graphs is studied through the
eigenvalues/eigenvectors of matrices associated with them. Many researchers
around the world, motivated by this theory, have defined some matrices
associated with hypergraph, aiming to develop a spectral hypergraph theory [8,
23, 24]. In 2012, Cooper and Dutle [6] proposed the study of hypergraphs by
means of the adjacency tensor. It is known, however, that to obtain
eigenvalues of tensors has a high computational and theoretical cost [12].
Perhaps for this reason, recently, some authors have renewed the interest to
study matrix representations of hypergraphs, as for example in [1, 7, 16, 18,
19, 25, 26].

The study of molecular orbital energy levels of $\pi$-electrons in conjugated
hydrocarbons may be seen as one of the oldest applications of spectral graph
theory [17, 14]. In those studies, graphs were used to represent hydrocarbon
molecules and it was shown that an approximation of the total $\pi$-electron
energy may be computed from the eigenvalues of the graph. Based on this
chemical concept, in 1977 Gutman [10] defined graph energy. In 2007, Nikiforov
[20] extended the concept of graph energy to matrices. For a matrix
$\mathbf{M}$, its energy $\mathcal{E}(\mathbf{M})$, is defined as the sum of
its singular values.

Let $\mathcal{H}$ be a hypergraph and $\mathbf{A}$ its adjacency matrix. We
define here the energy of $\mathcal{H}$ as $\mathcal{E}(\mathbf{A})$ and
denote it by $\mathcal{E}(\mathcal{H})$.

The main purpose of this paper is to discuss this natural definition of
(adjacency) energy of a hypergraph and to understand whether known properties
of graph energy may be extended to the structure of hypergraphs.

We highlight some of the results we obtain. We start by studying a particular
class of hypergraphs and determine which hyperstar has highest and lowest
energy among this class. By understanding the operations _sum_ and _product_
of hypergraphs, we obtain restrictions on the values that
$\mathcal{E}(\mathcal{H})$ may have. In particular, we show that
$\mathcal{E}(\mathcal{H})$ is never an odd number. In addition, we determine
bounds for the variation of the energy of a hypergraph $\mathcal{H}$, when we
perform operations on its edges or vertices. Our main contribution is to
obtain several lower and upper bounds relating $\mathcal{E}(\mathcal{H})$ to
important structural and spectral parameters.

The remaining of the paper is organized as follows. In Section 2, we recall
some basic definitions about hypergraphs and matrices that will be used. In
Section 3, we study the energy of a hyperstar. In Section 4, we prove that the
energy of a hypergraph can never be an odd number. In sections 5 and 6, we
study what happens when we delete or divide an edge of a hypergraph. Finally,
in Section 7, we will obtain bounds for the energy of a hypergrap. These
bounds are related to important spectral and structural parameters, such as
Zagreb index and spectral radius.

##  2\. Preliminaries

In this section, we shall present some basic definitions about hypergraphs and
matrices, as well as terminology, notation and concepts that will be useful in
our proofs.

A hypergraph $\mathcal{H}=(V,E)$ is a pair composed by a set of vertices
$V(\mathcal{H})$ and a set of (hyper)edges $E(\mathcal{H})$, where each edge
is a subset of $V(\mathcal{H})$, with cardinality greater than or equal $2$.
The rank and the co-rank of a hypergraph are defined as the largest and
smallest cardinality of its edges, respectively. $\mathcal{H}$ is said to be a
$k$-uniform hypergraph (or a $k$-graph) for $k\geq 2$, if all edges have the
same cardinality $k$. Let $\mathcal{H}=(V,E)$ and
$\mathcal{H}^{\prime}=(V^{\prime},E^{\prime})$ be hypergraphs, if
$V^{\prime}\subseteq V$ and $E^{\prime}\subseteq E$, then
$\mathcal{H}^{\prime}$ is a subgraph of $\mathcal{H}$. The complete $k$-graph
$\mathcal{K}^{[k]}_{n}$ on $n$ vertices, is a hypergraph, such that any subset
of $k$ vertices is an edge. A hypergraph $\mathcal{H}$ is linear if each pair
of edges has at most one common vertex.

A multi-hypergraph $\mathcal{H}=(V,E)$ is a pair composed by a set of vertices
$V(\mathcal{H})$ and a multi-set of (multi-)edges $E(\mathcal{H})$, where each
edge is a subset of $V(\mathcal{H})$.

Notice that in a hypergraph the edges must have at least two vertices and
distinct edges cannot have exactly the same vertices, while in a multi-
hypergraph there may be edges containing exactly the same vertices.
Additionally, there could be edges with one or zero vertices.

Let $\mathcal{H}=(V,E)$ be a (multi-)hypergraph. The edge neighborhood of a
vertex $v\in V$, denoted by $E_{[v]}$, is the set of all edges that contains
$v$. The degree of a vertex $v\in V$, denoted by $d(v)$, is the number of
edges that contain $v$. More precisely, $\;d(v)=|E_{[v]}|$. A hypergraph is
$r$-regular if $d(v)=r$ for all $v\in V$. We define the maximum, minimum and
average degrees, respectively, as

$\Delta(\mathcal{H})=\max_{v\in
V}\\{d(v)\\},\quad\delta(\mathcal{H})=\min_{v\in V}\\{d(v)\\},\quad
d(\mathcal{H})=\frac{1}{|V|}\sum_{v\in V}d(v).$

Let $\alpha=\\{v_{i_{1}},\ldots,v_{i_{r}}\\}\subset V$ be a subset of
vertices. We define the degree of a set $\alpha$, as the number of edges that
contain simultaneously all vertices of $\alpha$ and denote it for $d(\alpha)$.

Let $\mathcal{H}$ be a hypergraph. A walk of length $l$ is a sequence of
vertices and edges $v_{0}e_{1}v_{1}e_{2}\ldots e_{l}v_{l}$ where $v_{i-1}$ and
$v_{i}$ are distinct vertices contained in $e_{i}$ for each $i=1,\ldots,l$. A
cycle is a walk where $v_{0}=v_{l}.$ The hypergraph is connected, if for each
pair of vertices $u,w$ there is a walk $v_{0}e_{1}v_{1}e_{2}\cdots e_{l}v_{l}$
where $u=v_{0}$ and $w=v_{l}$. Otherwise, the hypergraph is disconnected.

Let $\mathcal{H}=(V,E)$ be a (multi-)hypergraph with $n$ vertices. The
adjacency matrix $\mathbf{A}(\mathcal{H})=(a_{ij})$ is defined as a square
matrix with $n$ rows. For each pair of vertices $i,j\in V$, if $i=j$, then
$a_{ii}=0$, and if $i\neq j$, then $a_{ij}=d(\\{i,j\\})$. We denote its
characteristic polynomial by
$P_{\mathbf{A}}(\lambda)=\det(\lambda\mathbf{I}_{n}-\mathbf{A})$. Its
eigenvalues will be denoted by
$\lambda_{1}(\mathbf{A})\geq\cdots\geq\lambda_{n}(\mathbf{A})$. If
$\mathbf{x}$ is an eigenvector of $\lambda$, we call the pair
$(\lambda,\mathbf{x})$, as an eigenpair of $\mathbf{A}$. The spectral radius
$\rho(\mathbf{A})$, is the largest absolute value of its eigenvalues. The
(adjacency) energy of a hypergraph is the sum of the singular values of its
adjacency matrix. We notice that $\mathbf{A}$ is a real and symmetric square
matrix, then its energy is the sum of the absolute values of the eigenvalues,
in other words

$\mathcal{E}(\mathcal{H})=\sum_{i=1}^{n}|\lambda_{i}|.$

Let $\mathcal{H}=(V,E)$ be a hypergraph with $n$ vertices. For a non-empty
subset of vertices $\alpha=\\{v_{1},\ldots,v_{t}\\}\subset V$ and a vector
$\mathbf{x}=(x_{i})$ of dimension $n$, we denote
$x(\alpha)=x_{v_{1}}+\cdots+x_{v_{t}}$. So we have,

$(\mathbf{A}(\mathcal{H})\mathbf{x})_{u}=\sum_{e\in
E_{[u]}}x\left(e-\\{u\\}\right),\quad\forall u\in V(\mathcal{H}).$

##  3\. The energy of a hyperstar

In this section, we will obtain the energy of a hyperstar and determe which
hyperstar with $n$ vertices has the highest energy.

First of all, we will present the definition of a power graph, and obtain an
algebraic property of its eigenvalues.

######  Definition 3.1.

Let $\mathcal{G}=(V,E)$ be a graph and let $k\geq 2$ be an integer. We define
the power graph $\mathcal{G}^{k}$ as the $k$-graph with the following sets of
vertices and edges

$V(\mathcal{G}^{k})=V(\mathcal{G})\cup\left(\bigcup_{e\in
E(\mathcal{G})}\varsigma_{e}\right)\;\;\textrm{and}\;\;E(\mathcal{G}^{k})=\\{e\cup\varsigma_{e}\colon
e\in E(\mathcal{G})\\},$

where $\varsigma_{e}=\\{v_{1}^{e},\ldots,v_{k-2}^{e}\\}$ for each edge $e\in
E(\mathcal{G})$.

We may say that $\mathcal{G}^{k}$ is obtained from a base graph
$\mathcal{G}=(V,E)$, adding $k-2$ new vertices to each edge $e\in
E(\mathcal{G})$. For each edge $e\in E(\mathcal{G})$, we denote
$e^{k}=e\cup\varsigma_{e}\in E(\mathcal{G}^{k})$. The spectrum of this class
has already been studied, see for example [3, 5].

We define a _hyperstar_ as a power graph of a star.

######  Example 3.2.

The power graph $(S_{4})^{3}$ of the star $S_{4}$ is illustrated in Figure 1.

$v_{1}$$v_{2}$$v_{3}$$v_{4}$

$v_{1}$$v_{2}$$v_{3}$$v_{4}$

Figure 1. The power hypergraph $(S_{4})^{3}$.

######  Lemma 1.

Let $\mathcal{H}$ be a $k$-graph having two vertices $u$ and $v$ which are
contained exactly in the same edges. If $(\lambda,\mathbf{x})$ is an eigenpair
of $\mathbf{A}(\mathcal{H})$ with $\lambda\neq-d(u)$, then $x_{u}=x_{v}$.

###### Proof.

We observe that,

$(\lambda+d(u))x_{u}=\sum_{e\in E_{[u]}}x(e)=\sum_{e\in
E_{[v]}}x(e)=(\lambda+d(v))x_{v}.$

Since $d(u)=d(v)$ and $\lambda\neq-d(u)$, then the result follows. ∎

######  Theorem 2.

Let $S_{n}$ be the star on $n$ vertices. If $k\geq 2$ is an integer, then
spectrum of $(S_{n})^{k}$ is:

$\\{(-1)^{(n-1)(k-2)},\;(k-2)^{n-2},\;r^{+},\;r^{-}\\},$

where $r^{+}$ and $r^{-}$ are the roots of $x^{2}-(k-2)x-(n-1)(k-1)=0$.

###### Proof.

Let $e\in E(S_{n})$ be an edge, let $\\{u_{1},\ldots,u_{k-1}\\}$ be the
vertices of degree one in $e^{k}$ and $2\leq i\leq k-1$. We can construct the
following family of $k-2$ linearly independent vectors.

$\mathbf{x}^{i}=\begin{cases}(x^{i})_{u_{1}}=\;\;\,1,\\\
(x^{i})_{u_{i}}=-1,\\\ (x^{i})_{u}\;=\;\;\,0,\textrm{ for }u\in
V((S_{n})^{k})-\\{u_{1},u_{i}\\}.\end{cases}$

Repeating this construction for the other edges of $S_{n}$, we obtain
$(n-1)(k-2)$ linearly independent eigenvectors, associated with $\lambda=-1$.

Let $E(S_{n})=\\{e_{1},\ldots,e_{n-1}\\}$ and $2\leq j\leq n-1$. We can
construct the following family of $n-2$ linearly independent vectors.

$\mathbf{z}^{j}=\begin{cases}(z^{j})_{u}=\;\;\,1,\textrm{ if }u\textrm{ is a
vertex of degree one in }(e_{1})^{k},\\\ (z^{j})_{v}=-1,\textrm{ if }v\textrm{
is a vertex of degree one in }(e_{j})^{k},\\\ (z^{j})_{w}\;=\;\,0,\textrm{ if
}w\textrm{ is not a vertex of degree one in }(e_{1})^{k}\textrm{ or in
}(e_{j})^{k}.\end{cases}$

So, we have $n-2$ linearly independent eigenvectors, associated with
$\lambda=k-2$.

Let $(\lambda,\mathbf{x})$ an eigenpair of $\mathbf{A}((S_{n})^{k})$, let
$E(S_{n})=\\{e_{1},\ldots,e_{n-1}\\}$, let
$\\{u^{j}_{1},\ldots,u^{j}_{k-1}\\}$ be the vertices of degree one in
$(e_{j})^{k}$, and let $v$ be the vertex of degree greater than one in
$(S_{n})^{k}$. By Lemma 1 we know that $x_{u^{j}_{i}}=x_{u^{j}_{1}}$ for all
$i=2,\ldots,k-1$, so the system of eigenvalues can be writen as:

$\begin{cases}\lambda x_{u_{1}^{1}}=x_{v}+(k-2)x_{u_{1}^{1}}\\\ \lambda
x_{u_{1}^{2}}=x_{v}+(k-2)x_{u_{1}^{2}}\\\ \vdots\\\ \lambda
x_{u_{1}^{m}}=x_{v}+(k-2)x_{u_{1}^{n-1}}\\\ \lambda
x_{v}=(k-1)x_{u_{1}^{1}}+\cdots+(k-1)x_{u_{1}^{n-1}}\\\ \end{cases}$

Notice that, if $\lambda\neq k-2$, then
$x_{u_{1}^{1}}=\cdots=x_{u_{1}^{n-1}}$, so the system can be write as:

$\begin{cases}\lambda x_{u_{1}^{1}}=x_{v}+(k-2)x_{u_{1}^{1}}\\\ \lambda
x_{v}=(n-1)(k-1)x_{u_{1}^{1}}\\\ \end{cases}$

By the first equation we have that, $x_{v}=(\lambda-k+2)x_{u_{1}^{1}}$, so by
second equation we conclude that

$\lambda(\lambda-k+2)x_{u_{1}^{1}}=(n-1)(k-1)x_{u_{1}^{1}},$

so

$\lambda^{2}+\lambda(-k+2)-(n-1)(k-1)=0.$

Finally, to conclude the result just notice that the sum of the multiplicities
of the eigenvalues is $|V((S_{n})^{k})|=(n-1)(k-1)+1$. ∎

######  Corollary 3.

Let $S_{n}$ be the star on $n$ vertices. If $k\geq 2$ is an integer, then

$\mathcal{E}((S_{n})^{k})=(k-2)(2n-3)+\sqrt{(k-2)^{2}+4(n-1)(k-1)}.$

###### Proof.

We just notice that

$(n-1)(k-2)|-1|+(n-2)|k-2|=(k-2)(2n-3).$

Now, observe that

$r^{+}=\frac{(k-2)+\sqrt{(k-2)^{2}+4(n-1)(k-1)}}{2}\geq 0$

and

$r^{-}=\frac{(k-2)-\sqrt{(k-2)^{2}+4(n-1)(k-1)}}{2}\leq 0,$

so $|r^{+}|+|r^{-}|=r^{+}-r^{-}=\sqrt{(k-2)^{2}+4(n-1)(k-1)}.$ ∎

######  Corollary 4.

If $\mathcal{S}$ is a hyperstar with $t$ vertices, then

$2\sqrt{t-1}=\mathcal{E}(S_{t})\leq\mathcal{E}(\mathcal{S})\leq\mathcal{E}((S_{2})^{t})=2(t-1).$

###### Proof.

If $\mathcal{S}$ is a hyperstar, then there is $2\leq n\leq t$ and $2\leq
k\leq t$ such that $\mathcal{S}=(S_{n})^{k}$. In this way, we have that
$t=(n-1)(k-1)+1$, so $n=\frac{t-1}{k-1}+1$, therefore

$\mathcal{E}(\mathcal{S})=2(t-1)\left(\frac{k-2}{k-1}\right)-(k-2)+\sqrt{(k-2)^{2}+4(t-1)}.$

Consider the function $f:[2,t]\rightarrow\mathbb{R},$ defined by

$f(x)=2(t-1)\left(\frac{x-2}{x-1}\right)-(x-2)+\sqrt{(x-2)^{2}+4(t-1)}.$

Computing its derivatives, we obtain

$\displaystyle f^{\prime}(x)$ $\displaystyle=$
$\displaystyle\frac{2(t-1)}{(x-1)^{2}}-1+\frac{x-2}{\sqrt{(x-2)^{2}+4(t-1)}},$
$\displaystyle f^{\prime\prime}(x)$ $\displaystyle=$
$\displaystyle\frac{-4(t-1)}{(x-1)^{3}}+\frac{4(t-1)}{((x-2)^{2}+4(t-1))^{\frac{3}{2}}}.$

We observe that
$f^{\prime\prime}(x)<0\;\Leftrightarrow\;(x-1)^{2}<(x-2)^{2}+4(t-1)\;\Leftrightarrow\;2x+1<4t,$
therefore $f^{\prime}(x)$ is a decreasing function. Now, we notice that
$f^{\prime}(t)=\frac{2}{t-1}-\frac{2}{t}>0$, so $f^{\prime}(x)>0$ for all
$x\in[2,t]$, so $f(x)$ is an increasing function and therefore the result
follows. ∎

##  4\. The energy of a hypergraph is never odd

In this section, we will prove that the energy of a hypergraph can never be an
odd number. More precisely, we prove that the energy of a hypergraph can not
be a $p$-th root of an odd number, and if it is a $p$-root of an even number
then its even prime factors have power smaller than $p$. Similar results have
already been obtained for graphs in [2, 22].

We start by defining some operations between hypergraphs, and computing the
spectrum of the resulting hypergraph.

######  Definition 4.1.

Let $\mathcal{H}$ and $\mathcal{G}$ be $k$-graphs. We define its sum
$\mathcal{H}\oplus\mathcal{G}$ as the $k$-graph, with the sets of vertices
$\;V(\mathcal{H}\oplus\mathcal{G})=V(\mathcal{H})\times V(\mathcal{G})$ and
edges

$\;E(\mathcal{H}\oplus\mathcal{G})=\\{\\{v\\}\times e:\;v\in
V(\mathcal{H}),\;e\in E(\mathcal{G})\\}\cup\\{f\times\\{u\\}:\;\;f\in
E(\mathcal{H}),u\in V(\mathcal{G})\\}.$

######  Example 4.2.

Let $\mathcal{H}$ and $\mathcal{G}$ be $3$-graphs, such that
$V(\mathcal{H})=\\{1,2,3,4\\}$, $E(\mathcal{H})=\\{123,234\\}$ and
$V(\mathcal{G})=\\{a,b,c\\}$, $E(\mathcal{G})=\\{abc\\}$. The sum
$\mathcal{H}\oplus\mathcal{G}$ has the following sets of vertices and edges

$V(\mathcal{H}\oplus\mathcal{G})=\\{(1,a),(1,b),(1,c),(2,a),(2,b),(2,c),(3,a),(3,b),(3,c),(4,a),(4,b),(4,c)\\}.$
$E(\mathcal{H}\oplus\mathcal{G})=\left\\{\begin{matrix}\\{(1,a),(1,b),(1,c)\\},&\\{(1,a),(2,a),(3,a)\\},\\\
\\{(2,a),(2,b),(2,c)\\},&\\{(1,b),(2,b),(3,b)\\},\\\
\\{(3,a),(3,b),(3,c)\\},&\\{(1,c),(2,c),(3,c)\\},\\\
\\{(4,a),(4,b),(4,c)\\},&\\{(2,a),(3,a),(4,a)\\},\\\
\\{(2,b),(3,b),(4,b)\\},&\\{(2,c),(3,c),(4,c)\\}\end{matrix}\right\\}_{.}$

######  Proposition 5.

If $\mathcal{G}$ and $\mathcal{H}$ are two $k$-graphs having eigenvalues $\mu$
and $\lambda$ with multiplicities $m_{1}$ and $m_{2}$, respectively, then
$\mu+\lambda$ is an eigenvalue of $\mathbf{A}(\mathcal{G}\oplus\mathcal{H})$,
with multiplicity $m_{1}\cdot m_{2}$.

###### Proof.

Suppose $\mathbf{x}$ an eigenvector of $\lambda$ in $\mathbf{A}(\mathcal{H})$
and $\mathbf{y}$ an eigenvector of $\mu$ in $\mathbf{A}(\mathcal{G})$.
Consider $(v,u)$ a vertex of $\mathcal{G}\oplus\mathcal{H}$, define a vector
$\mathbf{z}$ by $z_{(v,u)}=y_{v}x_{u}$. Thus,

$(\mathbf{A}\mathbf{z})_{(v,u)}=\\!\\!\\!\sum_{\alpha\in
E_{[(v,u)]}}\\!\\!\\!z(\alpha-(v,u))=\sum_{e\in E_{[u]}}y_{v}x(e-u)+\sum_{a\in
E_{[v]}}y(a-v)x_{u}=(\mu+\lambda)y_{v}x_{u}.$

Therefore, the result follows. ∎

######  Definition 4.3.

Let $\mathcal{H}$ and $\mathcal{G}$ be $k$-graphs.

  1. (1)

For each edge $e\in E(\mathcal{H})$, we say that a sequence
$\alpha=(v_{1},v_{2},\ldots,v_{k})$ is an ordered edge from $e$, if the set of
its elements is equal to the edge $e$.

  2. (2)

For each edge $e\in E(\mathcal{H})$, we denote by $S_{\mathcal{H}}(e)$ the set
of all ordered edges from $e$.

  3. (3)

Let $e\in E(\mathcal{H})$ and $f\in E(\mathcal{G})$. For
$\alpha=(v_{1},\ldots,v_{k})\in S_{\mathcal{H}}(e)$ and
$\beta=(u_{1},\ldots,u_{k})\in S_{\mathcal{G}}(f)$, we define its product as
the following set of ordered pairs

$\alpha\otimes\beta=\\{(v_{1},u_{1}),\ldots,(v_{k},u_{k})\\}.$

  4. (4)

We define the product $\mathcal{H}\otimes\mathcal{G}$ as the $k$-graph, with
the sets of vertices $\;V(\mathcal{H}\otimes\mathcal{G})=V(\mathcal{H})\times
V(\mathcal{G})$ and edges

$\;E(\mathcal{H}\otimes\mathcal{G})=\\{\alpha\otimes\beta:\;\alpha\in
S_{\mathcal{H}}(e),\textrm{ where }e\in E(\mathcal{H}),\textrm{ and }\beta\in
S_{\mathcal{G}}(f),\textrm{ where }f\in E(\mathcal{G}).\\}$

######  Example 4.4.

Let $\mathcal{H}$ and $\mathcal{G}$ be $3$-graphs, such that
$V(\mathcal{H})=\\{1,2,3,4\\}$, $E(\mathcal{H})=\\{123,234\\}$ and
$V(\mathcal{G})=\\{a,b,c\\}$, $E(\mathcal{G})=\\{abc\\}$. The product
$\mathcal{H}\otimes\mathcal{G}$ has the following sets of vertices and edges

$V(\mathcal{H}\otimes\mathcal{G})=\\{(1,a),(1,b),(1,c),(2,a),(2,b),(2,c),(3,a),(3,b),(3,c),(4,a),(4,b),(4,c)\\}.$
$E(\mathcal{H}\otimes\mathcal{G})=\left\\{\begin{matrix}\\{(1,a),(2,b),(3,c)\\},&\\{(2,a),(3,b),(4,c)\\},\\\
\\{(1,a),(2,c),(3,b)\\},&\\{(2,a),(3,c),(4,b)\\},\\\
\\{(1,b),(2,a),(3,c)\\},&\\{(2,b),(3,a),(4,c)\\},\\\
\\{(1,b),(2,c),(3,a)\\},&\\{(2,b),(3,c),(4,a)\\},\\\
\\{(1,c),(2,a),(3,b)\\},&\\{(2,c),(3,a),(4,b)\\},\\\
\\{(1,c),(2,b),(3,a)\\},&\\{(2,c),(3,b),(4,a)\\}\end{matrix}\right\\}_{.}$

######  Proposition 6.

If $\mathcal{G}$ and $\mathcal{H}$ are two $k$-graphs, with eigenvalues $\mu$
of multiplicity $m_{1}$ and $\lambda$ of multiplicity $m_{2}$ respectively,
then $\mu\lambda$ is an eigenvalue of
$\mathbf{A}(\mathcal{G}\otimes\mathcal{H})$, with multiplicity $m_{1}\cdot
m_{2}$.

###### Proof.

Suppose $\mathbf{x}$ an eigenvector of $\lambda$ in $\mathbf{A}(\mathcal{H})$
and $\mathbf{y}$ an eigenvector of $\mu$ in $\mathbf{A}(\mathcal{G})$.
Consider $(v,u)$ a vertex of $\mathcal{G}\otimes\mathcal{H}$, define a vector
$\mathbf{z}$ by $z_{(v,u)}=y_{v}x_{u}$. Thus,

$(\mathbf{A}\mathbf{z})_{(v,u)}=\\!\\!\\!\sum_{\alpha\in
E_{[(v,u)]}}\\!\\!\\!z(\alpha-(v,u))=\sum_{a\in E_{[v]}}\sum_{e\in
E_{[u]}}y(a-v)x(e-u)=\mu\lambda y_{v}x_{u}.$

Therefore, the result is true. ∎

######  Lemma 7.

Let $\mathcal{H}$ be a $k$-graph. If $\lambda_{1},\ldots,\lambda_{t}$ are all
positive eigenvalues of $\mathbf{A}(\mathcal{H})$, then
$\mathcal{E}(\mathcal{H})=2\sum_{i=1}^{t}\lambda_{i}$

###### Proof.

This follows, because the trace of the adjacency matrix is zero and equals the
sum of the eigenvalues. ∎

######  Remark 4.5.

We notice that the characteristic polynomial of $\mathcal{H}$ is a monic
polynomial with integer coefficients. Thus, if an eigenvalue of $\mathcal{H}$
is a rational number, then it has to be an integer.

######  Theorem 8.

Let $\mathcal{H}$ be a $k$-graph. If $\mathcal{E}(\mathcal{H})$ is a rational
number, then $\mathcal{E}(\mathcal{H})$ is even.

###### Proof.

Let $\lambda_{1},\ldots,\lambda_{t}$ be all positive eigenvalues of
$\mathcal{H}$, so $\mathcal{E}(\mathcal{H})=2\sum_{i=1}^{t}\lambda_{i}$, that
is $\sum_{i=1}^{t}\lambda_{i}$ is a rational number. By Proposition 5, we have
that $\sum_{i=1}^{t}\lambda_{i}$ is an eigenvalue of a hypergraph and by
Remark 4.5 we conclude that it must be an integer, therefore
$\mathcal{E}(\mathcal{H})=2\sum_{i=1}^{t}\lambda_{i}$ must be an even number.
∎

######  Theorem 9.

Let $p$ and $q$ be integers such that $p\geq 1$ and $0\leq q\leq p-1$ and $t$
be an odd integer. If $\mathcal{H}$ is a $k$-graph, then
$\mathcal{E}(\mathcal{H})\neq\sqrt[p]{2^{q}t}$.

###### Proof.

Suppose by contradiction that $\mathcal{E}(\mathcal{H})=\sqrt[p]{2^{q}t}$. In
this way, we have

$\left(\mathcal{E}(\mathcal{H})\right)^{p}=\left(2\sum_{i=1}^{t}\lambda_{i}\right)^{p}=2^{q}t\Rightarrow\left(\sum_{i=1}^{t}\lambda_{i}\right)^{p}=\frac{t}{2^{p-q}}.$

By Proposition 5 and 6, we have that
$\left(\sum_{i=1}^{t}\lambda_{i}\right)^{p}$ is an eigenvalue of a hypergraph,
however it is a non integral rational number, which is a contradiction. ∎

##  5\. Vertex and edge deletion

In this section we will study the variation of the energy of a hypergraph when
we delete a vertex or an edge of the hypergraph.

######  Definition 5.1.

Let $\mathcal{H}=(V,E)$ be a hypergraph, $v\in V$ be a vertex and
$e_{1},\ldots,e_{d}$ be all edges containing $v$, we define $\mathcal{H}-v$ by
$V(\mathcal{H}-v)=V(\mathcal{H})-\\{v\\}$ and

$E(\mathcal{H}-v)=(E(\mathcal{H})-\\{e_{1},\ldots,e_{d}\\})\cup\\{e_{1}-\\{v\\},\ldots,e_{d}-\\{v\\}\\}.$

######  Theorem 10.

Let $\mathcal{H}$ be a hypergraph and $v\in V(\mathcal{H})$ be a vertex, then

$\mathcal{E}(\mathcal{H})\geq\mathcal{E}(\mathcal{H}-v).$

###### Proof.

First notice that, $\mathbf{A}(\mathcal{H}-v)$ is a principal sub-matrix of
$\mathbf{A}(\mathcal{H})$. Let $\mu_{1},\ldots,\mu_{t}$ be the positive
eigenvalues of $\mathbf{A}(\mathcal{H}-v)$ and let
$\mu_{t+1},\ldots,\mu_{n-1}$ be the non positive eigenvalues of
$\mathbf{A}(\mathcal{H}-v)$. From the Interlace Theorem (see section 6.4 in
[9]), we have that

$\lambda_{1}(\mathcal{H})\geq\mu_{1},\quad\lambda_{2}(\mathcal{H})\geq\mu_{2},\cdots,\lambda_{t}(\mathcal{H})\geq\mu_{t}.$
$|\lambda_{n}(\mathcal{H})|\geq|\mu_{n-1}|,\quad|\lambda_{n-1}(\mathcal{H})|\geq|\mu_{n-2}|,\cdots,|\lambda_{t+2}(\mathcal{H})|\geq|\mu_{t+1}|.$

Therefore, $\mathcal{E}(\mathcal{H})\geq\mathcal{E}(\mathcal{H}-v).$ ∎

Now, we will bound the variation in energy when we delete an edge in the
hypergraph.

The energy of a matrix $\mathbf{M}$ is a generalization of the energy of
graphs, it is computed as the sum of its singular values. In particular, if
$\mathbf{M}$ is a real and symmetric square matrix, its energy is also the sum
of the absolute values of the eigenvalues.

######  Lemma 11 (Lemma 2.21, [21]).

If $\mathbf{M}$ and $\mathbf{N}$ are square matrices, then

$\mathcal{E}(\mathbf{M}+\mathbf{N})\leq\mathcal{E}(\mathbf{M})+\mathcal{E}(\mathbf{N}),\quad|\mathcal{E}(\mathbf{M})-\mathcal{E}(\mathbf{N})|\leq\mathcal{E}(\mathbf{M}-\mathbf{N}).$

######  Definition 5.2.

Let $\mathcal{H}=(V,E)$ be a hypergraph and $e\in E$ be an edge, we define
$\mathcal{H}-e=(V,E-\\{e\\})$.

######  Theorem 12.

Let $\mathcal{H}$ be a hypergraph and $e\in E(\mathcal{H})$ be an edge, then

$|\mathcal{E}(\mathcal{H})-\mathcal{E}(\mathcal{H}-e)|\leq 2|e|-2.$

###### Proof.

First, we observe that

$\displaystyle|\mathcal{E}(\mathcal{H})-\mathcal{E}(\mathcal{H}-e)|$
$\displaystyle\leq$
$\displaystyle\mathcal{E}\left(\mathbf{A}(\mathcal{H})-\mathbf{A}(\mathcal{H}-e)\right).$

The inequality above follows from Lemma 11. Now, we observe that

$\mathbf{M}:=\mathbf{A}(\mathcal{H})-\mathbf{A}(\mathcal{H}-e)=\left[\begin{array}[]{cccc|cccc}0&1&\cdots&1&0&\cdots&0\\\
1&0&\cdots&1&0&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots\\\
1&1&\cdots&0&0&\cdots&0\\\ \hline\cr 0&0&\cdots&0&0&\cdots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&0&0&\cdots&0\\\
\end{array}\right].$

That is, the eigenvalues of $\mathbf{M}$, are $|e|-1$ with multiplicity $1$
and $-1$ with multiplicity $|e|-1$, thus the energy of this matrix is

$\mathcal{E}\left(\mathbf{A}(\mathcal{H})-\mathbf{A}(\mathcal{H}-e)\right)=|e|-1+(|e|-1)|-1|=2|e|-2.$

Therefore, the result follows. ∎

######  Example 5.3.

Removing a vertex always implies that
$\mathcal{E}(\mathcal{H})\geq\mathcal{E}(\mathcal{H}-v).$ For edge deletion,
this is no longer true as we see bellow.

  1. (1)

$\mathcal{E}(\mathcal{H})>\mathcal{E}(\mathcal{H}-e)$ removing any edge $e\in
E$:
Let $\mathcal{K}_{5}^{[3]}$ be the complete $3$-uniform hypergraph with $5$
vertices and let $e\in E$. Notice that
$\mathcal{E}(K_{5}^{[3]})=24,\quad\textrm{and}\quad\mathcal{E}(K_{5}^{[3]}-e)=21.731.$

  2. (2)

$\mathcal{E}(\mathcal{H})<\mathcal{E}(\mathcal{H}-e)$ removing any edge $e\in
E$:
Let $\mathcal{H}=(V,E)$ be the hypergraph given by
$V(\mathcal{H})=\\{1,2,3,4,5,6\\}$ and
$E=\\{135,136,145,146,235,236,245,246\\}.$ We have that
$\mathcal{E}(\mathcal{H})=16.$ For $e\in E$, we have
$\mathcal{E}(\mathcal{H}-e)=16,4926.$

  3. (3)

$\mathcal{E}(\mathcal{H})=\mathcal{E}(\mathcal{H}-e)$ removing removing a
particular edge $e$:
Let $\mathcal{H}=(V,E)$ be the hypergraph give by two copies of
$\mathcal{K}_{5}^{[3]}$ with $3$ distinct edges of cardinality $2$ with no
common vertices connecting the two copies. Taking $e\in E$ as one of those
edges, we have that $\mathcal{E}(\mathcal{H})=\mathcal{E}(\mathcal{H}-e)$.

It is known that for graphs the same is true, that is, the energy of the graph
when removing an edge, can increase, decrease or remain the same, [15].

##  6\. Edge division

Some important concepts of graph theory can be generalized in more than one
way. In particular, removing edges of a graph can be generalized, in the
context of hypergraphs, as the deletion of edges (seen in the previous
section) or as the division of edges, that we will define in this section.

######  Definition 6.1.

Let $\mathcal{H}=(V,E)$ be a hypergraph and $F=\\{e_{1},\ldots,e_{t}\\}\subset
E$ a subset of edges. If we can divide each edge $e_{i}=e_{i}^{\prime}\cup
e_{i}^{\prime\prime}$ so that $|e_{i}^{\prime}|>0$, $|e_{i}^{\prime\prime}|>0$
and $e_{i}^{\prime}\cap e_{i}^{\prime\prime}=\emptyset$, we define the
following multi-hypergraph

$\mathcal{H}\triangleleft(F,F^{\prime},F^{\prime\prime}):=\big{(}V,(E\cup
F^{\prime}\cup F^{\prime\prime})-F\big{)},$

where $F^{\prime}=\\{e_{1}^{\prime},\ldots,e_{t}^{\prime}\\}$ and
$F^{\prime\prime}=\\{e_{1}^{\prime\prime},\ldots,e_{t}^{\prime\prime}\\}$.
Under these conditions, we say that the multi-hypergraph
$\mathcal{H}\triangleleft(F,F^{\prime},F^{\prime\prime})$ is obtained from
$\mathcal{H}$ by dividing the edges in $F$. When is not necessary to
explicitly say how the edges in $F$ are divided, we will denote simply by
$\mathcal{H}\triangleleft F$. If $F$ is an unitary set, $F=\\{e\\}$, we denote
by $\mathcal{H}\triangleleft e$.

######  Example 6.2.

Consider the hypergraph $\mathcal{H}$ with the following vertices set
$V(\mathcal{H})=\\{0,\ldots 9\\}$ and edges
$E(\mathcal{H})=\\{012,234567,789\\}$, in the Figure 2 we represent the
hypergraph $\mathcal{H}\triangleleft(\\{234567\\},\\{2345\\},\\{67\\})$.

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$

Figure 2. Edge division

######  Remark 6.3.

Let $\mathbf{J}$ be the matrix with all entries equal to $1$ and consider the
$n\times n$ block matrix given by
$\mathbf{M}=\left[\begin{array}[]{cc}\mathbf{0}_{p}&\mathbf{J}_{p\times q}\\\
\mathbf{J}_{q\times p}&\mathbf{0}_{q}\\\ \end{array}\right],$ where $p+q=n.$
$\mathbf{M}$ has only two non zero eigenvalues which are given by $\sqrt{pq}$
and $-\sqrt{pq}.$

######  Theorem 13.

Let $\mathcal{H}$ be a hypergraph, $e\in E(\mathcal{H})$ an edge and
$e=e^{\prime}\cup e^{\prime\prime}$ a division of this edge. Then

$|\mathcal{E}(\mathcal{H})-\mathcal{E}(\mathcal{H}\triangleleft e)|\leq
2\sqrt{|e^{\prime}||e^{\prime\prime}|}.$

###### Proof.

First, notice that

$|\mathcal{E}(\mathcal{H})-\mathcal{E}(\mathcal{H}\triangleleft
e)|\leq\mathcal{E}(\mathbf{A}(\mathcal{H})-\mathbf{A}(\mathcal{H}\triangleleft
e)).$

We can assume that $e=\\{1,2,\ldots r\\}$, where
$r=|e^{\prime}|+|e^{\prime\prime}|.$ Now, also notice that

$\mathbf{A}(\mathcal{H})-\mathbf{A}(\mathcal{H}\triangleleft
e)=\left[\begin{array}[]{ccc}\mathbf{0}_{|e^{\prime}|}&\mathbf{J}_{|e^{\prime}|\times|e^{\prime\prime}|}&0\\\
\mathbf{J}_{|e^{\prime\prime}|\times|e^{\prime}|}&\mathbf{0}_{|e^{\prime\prime}|}&0\\\
0&0&0\\\ \end{array}\right].$

From Remark 6.3, we have that
$\mathcal{E}(\mathbf{A}(\mathcal{H})-\mathbf{A}(\mathcal{H}\triangleleft
e))=2\sqrt{|e^{\prime}||e^{\prime\prime}|}.$ ∎

######  Lemma 14.

Let $\mathcal{H}$ be a hypergraph and $e\in E(\mathcal{H})$ an isolated edge
(that is, the intersection of $e$ with any other edge is empty), then
$|\mathcal{E}(\mathcal{H})-\mathcal{E}(\mathcal{H}\triangleleft e)|=2.$

###### Proof.

We can assume $\mathcal{H}=\mathcal{H}^{\prime}\cup e$, where $e=\\{1,2,\ldots
r\\}$ and $r=|e^{\prime}|+|e^{\prime\prime}|.$ We have that

$\mathbf{A}(\mathcal{H})=\left[\begin{array}[]{cc}(\mathbf{J}-\mathbf{I})_{|e|\times|e|}&\mathbf{0}\\\
\mathbf{0}&\mathbf{A}(\mathcal{H}^{\prime})\\\ \end{array}\right]\quad
and\quad\mathbf{A}(\mathcal{H}\triangleleft
e)=\left[\begin{array}[]{ccc}(\mathbf{J}-\mathbf{I})_{|e^{\prime}|\times|e^{\prime}|}&\mathbf{0}&\mathbf{0}\\\
\mathbf{0}&(\mathbf{J}-\mathbf{I})_{|e^{\prime\prime}|\times|e^{\prime\prime}|}&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}&\mathbf{A}(\mathcal{H}^{\prime})\\\ \end{array}\right].$

Therefore

$\mathcal{E}(\mathcal{H})=\mathcal{E}(\mathcal{H}^{\prime})+2|e|-2\quad
and\quad\mathcal{E}(\mathcal{H}\triangleleft
e)=\mathcal{E}(\mathcal{H}^{\prime})+2|e^{\prime}|-2+2|e^{\prime\prime}|-2.$

So

$\mathcal{E}(\mathcal{H})-\mathcal{E}(\mathcal{H}\triangleleft
e)=2|e|-2-2(|e^{\prime}|+|e^{\prime\prime}|)+4=2.$

∎

######  Remark 6.4.

Let $\mathcal{H}$ be a hypergraph and $e\in E(\mathcal{H})$ an isolated edge,
then the equality
$|\mathcal{E}(\mathcal{H})-\mathcal{E}(\mathcal{H}\triangleleft
e)|=2\sqrt{|e^{\prime}||e^{\prime\prime}|}$ holds if, and only if, $|e|=2.$

######  Definition 6.5.

Let $\mathcal{H}=(V,E)$ be a connected hypergraph and $F\subset E$ a subset of
edges. The ordered triple $(F,F^{\prime},F^{\prime\prime})$ is said to be a
weak cut, if $\mathcal{H}\triangleleft(F,F^{\prime},F^{\prime\prime})$ is
disconnected, and for no proper subset $P\subset F$, the multi-hypergraph
$\mathcal{H}\triangleleft P$ is disconnected, considering any way to divide
the edges of $P$.

For our next result we need the following theorem from [15].

######  Theorem 15.

[15] For a real and symmetric partitioned matrix
$\mathbf{C}=\left[\begin{matrix}\mathbf{M}&\mathbf{X}\\\
\mathbf{Y}&\mathbf{B}\end{matrix}\right]$ where both $\mathbf{M}$ and
$\mathbf{B}$ are square matrices, we have

$\sum_{j}|\lambda_{j}(\mathbf{M})|+\sum_{j}|\lambda_{j}(\mathbf{B})|\leq\sum_{j}|\lambda_{j}(\mathbf{C})|.$

Moreover, equality holds if and only if there exist unitary matrices
$\mathbf{U}$ and $\mathbf{V}$ such that
$\left[\begin{matrix}\mathbf{UM}&\mathbf{UX}\\\
\mathbf{VY}&\mathbf{VB}\end{matrix}\right]$ is positive semi-definite.

######  Theorem 16.

Let $\mathcal{H}=(V,E)$ be a connected hypergraph. If $F\subset E$ is a weak
cut, then $\mathcal{E}(\mathcal{H}\triangleleft
F)\leq\mathcal{E}(\mathcal{H})$.

###### Proof.

Notice that $\mathcal{H}\triangleleft F$ is the disjoint union of two multi-
hypergraphs, lets say $\mathcal{H}\triangleleft
F=\mathcal{H}_{1}\cup\mathcal{H}_{2}$. So we have that

$\mathbf{A}(\mathcal{H}\triangleleft
F)=\left[\begin{matrix}\mathbf{A}(\mathcal{H}_{1})&\mathbf{0}\\\
\mathbf{0}&\mathbf{A}(\mathcal{H}_{1})\end{matrix}\right]\quad\textrm{ and
}\quad\mathbf{A}(\mathcal{H})=\left[\begin{matrix}\mathbf{A}(\mathcal{H}_{1})&\mathbf{X}\\\
\mathbf{X}^{T}&\mathbf{A}(\mathcal{H}_{2})\end{matrix}\right],$

where $\mathbf{X}=(x_{ij})$ is a matrix of order
$|V(\mathcal{H}_{1})|\times|V(\mathcal{H}_{2})|$ and $x_{ij}$ is the number of
edges in $F$ that contain the vertices $i\in V(\mathcal{H}_{1})$ e $j\in
V(\mathcal{H}_{2})$.

Therefore from Theorem 15 , we have that
$\mathcal{E}(\mathcal{H})\geq\mathcal{E}(\mathcal{H}_{1})+\mathcal{E}(\mathcal{H}_{2})=\mathcal{E}(\mathcal{H}\triangleleft
F).$ ∎

######  Lemma 17 (Exercise 2 of Section 7.1, [13]).

A positive semidefinite matrix has a zero entry on its main diagonal if and
only if the entire row and column to which that entry belongs is zero.

######  Theorem 18.

Let $\mathcal{H}=(V,E)$ be a connected hypergraph, $e\in E$ an edge and $v\in
e$ a vertex. If $(e,e-\\{v\\},\\{v\\})$ is a weak cut, then
$\mathcal{E}(\mathcal{H}\triangleleft(e,e-\\{v\\},\\{v\\}))<\mathcal{E}(\mathcal{H})$.

###### Proof.

Notice that
$\mathcal{H}\triangleleft(e,e-\\{v\\},\\{v\\})=\mathcal{H}_{1}\cup\mathcal{H}_{2}$.
So we have that

$\mathbf{A}(\mathcal{H}\triangleleft(e,e-\\{v\\},\\{v\\}))=\left[\begin{matrix}\mathbf{A}(\mathcal{H}_{1})&\mathbf{0}\\\
\mathbf{0}&\mathbf{A}(\mathcal{H}_{1})\end{matrix}\right]\quad\textrm{ and
}\quad\mathbf{A}(\mathcal{H})=\left[\begin{matrix}\mathbf{A}(\mathcal{H}_{1})&\mathbf{X}\\\
\mathbf{X}^{T}&\mathbf{A}(\mathcal{H}_{2})\end{matrix}\right],$

with $x_{ij}=1$ if $i\in e-\\{v\\}$ and $j=v$, and with $x_{ij}=0$ otherwise.
We can label the vertices in such way that the first row of $\mathbf{X}$ is
the only non null row.

Supposing that
$\mathcal{E}(\mathcal{H}\triangleleft(e,e-\\{v\\},\\{v\\}))=\mathcal{E}(\mathcal{H})$,
from Theorem 15 we know that exist orthogonal matrices $\mathbf{U}$ and
$\mathbf{W}$, such that

$\left[\begin{matrix}\mathbf{U}\mathbf{A}(\mathcal{H}_{1})&\mathbf{U}\mathbf{X}\\\
\mathbf{W}\mathbf{X}^{T}&\mathbf{W}\mathbf{A}(\mathcal{H}_{1})\end{matrix}\right],$

is a positive semidefined matrix. So we have that
$(\mathbf{U}\mathbf{X})^{T}=\mathbf{W}\mathbf{X}^{T}$, because of the
structure of $\mathbf{X}$ we have that
$\mathbf{W}=\left[\begin{matrix}\alpha&\mathbf{0}\\\
\mathbf{0}&\mathbf{W}_{1}\end{matrix}\right]$, with $|\alpha|=1$ and
$\mathbf{W}_{1}$ is a orthogonal matrix. So, we can write
$\mathbf{A}(\mathcal{H}_{1})=\left[\begin{matrix}0&\mathbf{y}^{T}\\\
\mathbf{y}&\mathbf{A}_{1}\end{matrix}\right]$, therefore
$\mathbf{W}\mathbf{A}(\mathcal{H}_{1})=\left[\begin{matrix}0&\alpha\mathbf{y}^{T}\\\
\mathbf{W}_{1}\mathbf{y}&\mathbf{W}_{1}\mathbf{A}_{1}\end{matrix}\right]$.
From Lemma 17, we have that $\mathbf{y}=\mathbf{0}$, but this implies that $v$
has no neighbor in $\mathcal{H}_{2}$ what contradicts the fact that
$\mathcal{H}$ is connected. ∎

######  Definition 6.6.

A hypergraph $\mathcal{T}$ is a hypertree if $\mathcal{T}$ is connected and
has no cycles.

For a hypertree $\mathcal{T}$ we notice that:

  1. (1)

$\mathcal{T}$ is a linear hypergraph, since if there exist two edges $e_{1}$ e
$e_{2}$ that have two (or more) common vertices $u$ e $v$, then we can
construct the cycle $ue_{1}ve_{2}u$.

  2. (2)

for any edge $e$ of a hypertree, if $v\in e$, then $(e,e-\\{v\\},v)$ is a weak
cut.

######  Corollary 19.

Let $\mathcal{T}$ be a hypertree. If $e\in E$ is an edge and $v\in e$ is a
vertex, then
$\mathcal{E}(\mathcal{T}\triangleleft(e,e-\\{v\\},\\{v\\}))<\mathcal{E}(\mathcal{T})$.

######  Remark 6.7.

Let $\mathcal{H}=(V,E)$ be a connected hypergraph and $e\in E$ an edge. If
$v\in e$ has degree $1$, then $(e,e-\\{v\\},v)$ is a weak cut.

######  Corollary 20.

Let $\mathcal{H}=(V,E)$ be a connected hypergraph and $e\in E$ an edge. If
$v\in e$ has degree $1$, then
$\mathcal{E}(\mathcal{H}\triangleleft(e,e-\\{v\\},\\{v\\}))<\mathcal{E}(\mathcal{H})$.

######  Example 6.8.

As it happens in $\mathcal{H}-e$, when we divide edges we can also have
$\mathcal{E}(\mathcal{H})>\mathcal{E}(\mathcal{H}\triangleleft e)$ and
$\mathcal{E}(\mathcal{H})<\mathcal{E}(\mathcal{H}\triangleleft e)$.

  1. (1)

$\mathcal{E}(\mathcal{H})>\mathcal{E}(\mathcal{H}\triangleleft e)$, by doing
any division at any edge:
Let $\mathcal{K}_{5}^{[4]}$ be the complete $4$-uniform hypergraph with $5$
vertices and observe that $\mathcal{E}(\mathcal{K}_{5}^{[4]})=24.$ Let $e\in
E$. Without loss of generality, we can assume that $e=\\{1,2,3,4\\}.$ We can
divide $e$ in two different ways.

Case 1: If $e^{\prime}=\\{1,2\\}$ e $e"=\\{3,4\\}$, then
$\mathcal{E}(\mathcal{K}_{5}^{[4]}\triangleleft e)=20,8924.$

Case 2: If $e^{\prime}=\\{1\\}$ e $e"=\\{2,3,4\\}$, then
$\mathcal{E}(\mathcal{K}_{5}^{[4]}\triangleleft e)=21,7438.$

  2. (2)

$\mathcal{E}(\mathcal{H})<\mathcal{E}(\mathcal{H}\triangleleft e)$, by doing
any division at any edge:

Let $\mathcal{H}=(V,E)$ be the hypergraph give by $V=\\{1,2,3,4,5,6,7,8\\}$
and

$E=\left\\{\begin{matrix}1357,1358,1367,1368,1457,1458,1467,1468,\\\
2357,2358,2367,2368,2457,2458,2467,2468\end{matrix}\right\\}_{.}$

We have that $\mathcal{E}(\mathcal{H})=48$. Now, let $e\in E$ be any edge.
Without loss of generality, we can assume that $e=\\{1,3,5,7\\}.$ We can
divide $e$ in two different ways.

Case 1: If $e^{\prime}=\\{1,3\\}$ e $e"=\\{5,7\\}$, then
$\mathcal{E}(\mathcal{H})=48.3294$.

Case 2: If $e^{\prime}=\\{1\\}$ e $e"=\\{3,5,7\\}$, then
$\mathcal{E}(\mathcal{H})=48,4723$.

  3. (3)

$\mathcal{E}(\mathcal{H})=\mathcal{E}(\mathcal{H}\triangleleft e)$ by doing a
particular division at a particular edge:

Example 5.3 (3) also works on this case. Notice that the edge $e$ has
cardinality $2$ and in this case we have that
$\mathbf{A}(\mathcal{H}\triangleleft e)=\mathbf{A}(\mathcal{H}-e).$

##  7\. Sharp bounds for energy

In this section we will obtain bounds for the adjacency energy of a
hypergraph. These bounds will be computed as functions of important and well
known spectral and structural parameters.

######  Definition 7.1.

Let $\mathcal{H}$ be a hypergraph. Its Zagreb index is defined as the sum of
the squares of the degrees of its vertices. More precisely

$Z(\mathcal{H})=\sum_{v\in V(\mathcal{H})}d(v)^{2}.$

This is an important parameter in graph theory, having chemistry applications,
see the survey [11].

We will start by presenting some upper bounds for the energy of a hypergraph

######  Theorem 21.

Let $\mathcal{H}=(V,E)$ be a hypergraph with rank $r$. If $|E|=m$ and $|V|=n$,
then

$\mathcal{E}(\mathcal{H})\leq\sqrt{n(r-1)Z(\mathcal{H})}\leq\sqrt{nm(r^{2}-r)\Delta}.$

Equality holds if, and only if, $\mathcal{H}$ is a $2$-graph with isolated
edges and no isolated vertices or if $\mathcal{H}$ is an edgeless hypergraph.

###### Proof.

Let $\lambda_{1},\cdots,\lambda_{n}$ be the eigenvalues of
$\mathbf{A}(\mathcal{H})$. Then,

$\sum_{i=1}^{n}\lambda_{i}^{2}=\mathrm{Tr}(\mathbf{A}^{2})=\sum_{i=1}^{n}\sum_{j=1}^{n}d(\\{i,j\\})^{2}\stackrel{{\scriptstyle(*)}}{{\leq}}\sum_{i=1}^{n}\left(d(i)\sum_{j=1}^{n}d(\\{i,j\\})\right)\stackrel{{\scriptstyle(**)}}{{\leq}}(r-1)\sum_{i=1}^{n}d(i)^{2}.$

Notice that the equality $(*)$ holds only on hypergraphs with the following
property: If two vertices $u$ e $v$ are neighbors then they belong to the same
edges. But this only happens on hypergraphs made of isolated edges (and
possibly some isolated vertices). Also notice that the equality $(**)$ holds
only on uniform hypergraphs. Then,

$\sum_{i=1}^{n}\lambda_{i}^{2}\leq(r-1)Z(\mathcal{H}).$

Therefore,

$\mathcal{E}(\mathcal{H})^{2}=\left(\sum_{i=1}^{n}|\lambda_{i}|\right)^{2}\stackrel{{\scriptstyle(cs)}}{{\leq}}n\left(\sum_{i=1}^{n}\lambda_{i}^{2}\right)\leq
n(r-1)Z(\mathcal{H}).$

By Cauchy Schwarz inequality, we know that the equality $(cs)$ only is true
when all the eigenvalues have the same absolute values. For uniform hypegraphs
made of isolated edges, this only happens when there is no isolated vertex and
the edges have size $2$.

Since,

$Z(\mathcal{H})=\sum_{i=1}^{n}d(i)^{2}\leq\sum_{i=1}^{n}\Delta d(i)\leq\Delta
rm$

And this equality occurs only on regulars hypergraphs, we conclude that

$\mathcal{E}(\mathcal{H})\leq\sqrt{n(r-1)Z(\mathcal{H})}\leq\sqrt{nm(r^{2}-r)\Delta}$

With all the equalities happening only if $\mathcal{H}$ is a $2$-graph with
isolated edges and no isolated vertices or if $\mathcal{H}$ has no edges. ∎

######  Theorem 22.

Let $\mathcal{H}=(V,E)$ be a hypergraph with rank $r$ and $|V|=n$, then

$\mathcal{E}(\mathcal{H})\leq\lambda_{1}+\sqrt{(n-1)[(r-1)Z(\mathcal{H})-\lambda_{1}^{2}]}.$

Equality holds if, and only if, $\mathcal{H}$ is a $2$-graph with isolated
edges and no isolated vertices or if $\mathcal{H}$ is an edgeless hypergraph.

###### Proof.

From Theorem 21 we have that

$\lambda_{1}^{2}+\sum_{i=2}^{n}\lambda_{i}^{2}=\sum_{i=1}^{n}\lambda_{i}^{2}\leq(r-1)Z(\mathcal{H}).$

With equality holding only on uniform hypergraphs made of isolated edges and
possibly some isolated vertices.

Therefore,

$(\mathcal{E}(\mathcal{H})-\lambda_{1})^{2}=(\sum_{i=2}^{n}|\lambda_{i}|)^{2}\stackrel{{\scriptstyle(cs)}}{{\leq}}(n-1)\sum_{i=2}^{n}\lambda_{i}^{2}\leq(n-1)[(r-1)Z(\mathcal{H})-\lambda_{1}^{2}].$

Notice that the equality $(cs)$ only is true when
$|\lambda_{2}|=|\lambda_{3}|=\ldots=|\lambda_{n}|$. For uniform hypergraphs
made of isolated edges, this can only occur when all the edges have size 2 and
there is no isolated vertices. ∎

######  Lemma 23.

Let $\mathcal{H}$ be a hypergraph with rank $r$ and co-rank $s$, then

$(s-1)\delta\leq\lambda_{1}\leq(r-1)\Delta.$

###### Proof.

Let $\mathbf{x}=(x_{i})$ be the non negative eigenvector associated to the
eigenvalue $\lambda_{1}$. Taking vertices $v,u\in V$ such that $x_{v}$ has
maximum value and $x_{u}$ has minimum value, we have that

$\lambda_{1}x_{v}=(\mathbf{A}\mathbf{x})_{v}=\sum_{e\in
E_{[v]}}x\left(e-\\{v\\}\right)$

therefore

$\lambda_{1}=\sum_{e\in
E_{[v]}}\frac{x\left(e-\\{v\\}\right)}{x_{v}}\leq\Delta\frac{(r-1)x_{v}}{x_{v}}=(r-1)\Delta.$

The other inequality follows similarly, using $x_{u}$. ∎

######  Corollary 24.

Let $\mathcal{H}=(V,E)$ be a $r$-uniform and $d$-regular hypergraph, then

$\mathcal{E}(\mathcal{H})\leq(r-1)d+\sqrt{(n-1)[(r-1)nd^{2}-(r-1)^{2}d^{2}]}.$

Now we will obtain some lower bounds for the adjacency energy.

######  Lemma 25.

Let $\mathcal{H}=(V,E)$ be a hypergraph with $|V|=n$, then

$\mathcal{E}(\mathcal{H})^{2}\geq 2\sum_{i=1}^{n}\lambda_{i}^{2}.$

Equality holds only if $\mathcal{H}$ has at most one $2$-uniform complete
bipartite connected component and possibly isolated vertices, or if
$\mathcal{H}$ has no edges.

###### Proof.

First, notice that

$0=\left(\sum_{i=1}^{n}\lambda_{i}\right)^{2}=\sum_{i=1}^{n}\lambda_{i}^{2}+2\sum_{i<j}\lambda_{i}\lambda_{j}.$

Therefore,

$\sum_{i=1}^{n}\lambda_{i}^{2}=-2\sum_{i<j}\lambda_{i}\lambda_{j},\quad\Rightarrow\quad\sum_{i<j}|\lambda_{i}\lambda_{j}|\stackrel{{\scriptstyle(1)}}{{\geq}}\left|\sum_{i<j}\lambda_{i}\lambda_{j}\right|=\frac{1}{2}\sum_{i=1}^{n}\lambda_{i}^{2}.$

Then,

$\mathcal{E}(\mathcal{H})^{2}=\left(\sum_{i=1}^{n}|\lambda_{i}|\right)^{2}=\sum_{i=1}^{n}|\lambda_{i}|^{2}+2\sum_{i<j}|\lambda_{i}\lambda_{j}|\stackrel{{\scriptstyle(2)}}{{\geq}}2\sum_{i=1}^{n}\lambda_{i}^{2}.$

Notice that equality $(1)$ and consequently $(2)$ occur only if every non zero
eigenvalue have the same signal. If $\mathcal{H}$ has at least three non zero
eigenvalues (not necessarily distincts) that is impossible to happen.
Therefore $\mathcal{H}$ must have at most one 2-uniform complete bipartite
connected component and possibly isolated vertices. ∎

In what follows, we present three lower bounds for the energy of a hypergraph,
based on different parameters.

######  Theorem 26.

Let $\mathcal{H}=(V,E)$ be a hypergraph with co-rank $s$, $|V|=n$ and average
degree $d(\mathcal{H})$ then

$\mathcal{E}(\mathcal{H})\geq\sqrt{2n(s-1)d(\mathcal{H})}.$

Equality holds if, and only if, $\mathcal{H}$ is the $2$-uniform hypergraph
with at most one complete bipartite connected component.

###### Proof.

It suffices to note that

$\mathcal{E}(\mathcal{H})^{2}\geq
2\sum_{i=1}^{n}\lambda^{2}_{i}=2\sum_{i=1}^{n}\sum_{j=1}^{n}d(i,j)^{2}\stackrel{{\scriptstyle(3)}}{{\geq}}2\sum_{i=1}^{n}\sum_{j=1}^{n}d(i,j)\geq
2n(s-1)d(\mathcal{H}).$

∎

######  Theorem 27.

Let $\mathcal{H}=(V,E)$ be a hypergraph with co-rank $s$ and $|V|=n$, then

$\mathcal{E}(\mathcal{H})\geq\sqrt{\frac{2(s-1)^{2}}{n}Z(\mathcal{H})}.$

Equality holds if, and only if, $\mathcal{H}$ is the $2$-uniform hypergraph
with only one edge and no isolated vertices or if $\mathcal{H}$ has no edges.

###### Proof.

It suffices to note that

$\mathcal{E}(\mathcal{H})^{2}\geq
2\sum_{i=1}^{n}\sum_{j=1}^{n}d(i,j)^{2}\stackrel{{\scriptstyle(4)}}{{\geq}}2\sum_{i=1}^{n}\left(\frac{1}{n}\left(\sum_{j=1}^{n}d(i,j)\right)^{2}\right)\geq\frac{2(s-1)^{2}}{n}Z(\mathcal{H}).$

Notice that equality $(4)$ holds only if $d(i,j)$ is constant for each $i$.
The only complete bipartite graph with this property is the edge. ∎

######  Remark 7.2.

Let $\mathcal{H}$ be a hypergraph with $n$ vertices and co-rank $s$. We can
compare the bounds provided by these two theorems. We define
$b(\mathcal{H})=\sqrt{2n(s-1)d(\mathcal{H})}$ e
$B(\mathcal{H})=\sqrt{\frac{2(s-1)^{2}}{n}Z(\mathcal{H})}$. It is easy to see
that:

  1. (1)

If $\frac{\delta^{2}}{\Delta}\geq\frac{n}{s-1}$, then
$b(\mathcal{H})\leq\sqrt{2n(s-1)\Delta}\leq\sqrt{2(s-1)^{2}\delta^{2}}\leq
B(\mathcal{H}).$

  2. (2)

If $\frac{\Delta^{2}}{\delta}\leq\frac{n}{s-1}$, then
$B(\mathcal{H})\leq\sqrt{2(s-1)^{2}\Delta^{2}}\leq\sqrt{2n(s-1)\delta}\leq
b(\mathcal{H}).$

  3. (3)

In particular, if $\mathcal{H}$ is $d$-regular, then:

    1. (a)

If $d=\frac{n}{s-1}$, then $b(\mathcal{H})=B(\mathcal{H})$.

    2. (b)

If $d>\frac{n}{s-1}$, then $b(\mathcal{H})<B(\mathcal{H})$.

    3. (c)

If $d<\frac{n}{s-1}$, then $b(\mathcal{H})>B(\mathcal{H})$.

######  Theorem 28.

Let $\mathcal{H}$ be a hypergraph, then

$\mathcal{E}(\mathcal{H})\geq\sqrt{\frac{2n}{n-1}\lambda_{1}^{2}}.$

Equality holds if, and only if, $\mathcal{H}$ is the $2$-uniform hypergraph
with one edge and two vertices or if $\mathcal{H}$ has no edges.

###### Proof.

Notice that

$\sum_{i=1}^{n}\lambda_{i}^{2}=\lambda_{1}^{2}+\sum_{i=2}^{n}\lambda_{i}^{2}\stackrel{{\scriptstyle(cs)}}{{\geq}}\lambda_{1}^{2}+\frac{1}{n-1}(\sum_{i=2}^{n}\lambda_{i})^{2}=\lambda_{1}^{2}+\frac{(-\lambda_{1})^{2}}{n-1}=\frac{n}{n-1}\lambda_{1}^{2}.$

Therefore,

$\mathcal{E}(\mathcal{H})^{2}\geq
2\sum_{i=1}^{n}\lambda_{i}^{2}\geq\frac{2n}{n-1}\lambda_{1}^{2}.$

Equality $(cs)$ only is true when
$|\lambda_{2}|=|\lambda_{3}|=\ldots=|\lambda_{n}|$. For complete bipartite
graphs, this only holds when $\mathcal{H}=K_{2}.$ ∎

Note that the proofs of the previous theorems leads immediately to distinct
low bounds of $\sum_{i=1}^{n}\lambda_{i}^{2}$. More specifically, we have the
following lemma, which will be useful later.

######  Lemma 29.

Let $\mathcal{H}$ be a hypergraph with co-rank $s$ and $|V|=n.$

  1. (1)

$\sum_{i=1}^{n}\lambda_{i}^{2}\geq n(s-1)d(\mathcal{H}).$

  2. (2)

$\sum_{i=1}^{n}\lambda_{i}^{2}\geq\frac{(s-1)^{2}}{n}Z(\mathcal{H}).$

  3. (3)

$\sum_{i=1}^{n}\lambda_{i}^{2}\geq\frac{n}{n-1}\lambda_{1}^{2}.$

######  Lemma 30.

Let $\mathcal{H}=(V,E)$ be a hypergraph with $|V|=n$, then

$\mathcal{E}(\mathcal{H})^{2}\geq\sum_{i=1}^{n}\lambda_{i}^{2}+n(n-1)|\det(\mathbf{A})|^{\frac{2}{n}}.$

Equality holds if $|\lambda_{i}|=|\lambda_{j}|$ for every $i\neq j$, that is,
if $\mathcal{H}$ is the graph with one edge and two vertices .

###### Proof.

First, notice that

$\frac{1}{n(n-1)}\sum_{i\neq
j}|\lambda_{i}\lambda_{j}|\stackrel{{\scriptstyle(1)}}{{\geq}}\prod_{i\neq
j}|\lambda_{i}\lambda_{j}|^{\frac{1}{n(n-1)}}=\prod_{i=1}^{n}|\lambda_{i}|^{\frac{2}{n}}=|\det(\mathbf{A})|^{\frac{2}{n}}$

Therefore,

$\mathcal{E}(\mathcal{H})^{2}=(\sum_{i=1}^{n}|\lambda_{i}|)^{2}=\sum_{i=1}^{n}\lambda_{i}^{2}+\sum_{i\neq
j}|\lambda_{i}\lambda_{j}|\geq\sum_{i=1}^{n}\lambda_{i}^{2}+n(n-1)|\det(\mathbf{A})|^{\frac{2}{n}}.$

Where $(1)$ holds because the arithmetic mean of non negative numbers is
greater or equal than the geometric mean. That is:

$\frac{x_{1}+\ldots+x_{n}}{n}\geq(x_{1}\ldots x_{n})^{\frac{1}{n}}$

and equality holds if, and only if, $x_{1}=\ldots=x_{n}.$ ∎

From these lemmas, we establish other lower bounds for the energy of a
hypergraph, depending on the determinant of its adjacency matrix.

######  Theorem 31.

Let $\mathcal{H}=(V,E)$ be a hypergraph with co-rank $s$ and $|V|=n$, then

$\mathcal{E}(\mathcal{H})\geq\sqrt{n(s-1)d(\mathcal{H})+n(n-1)|\det(\mathbf{A})|^{\frac{2}{n}}}.$

###### Proof.

Use Lemma 30 and Remark 29 $(1)$. ∎

######  Theorem 32.

Let $\mathcal{H}=(V,E)$ be a hypergraph with co-rank $s$ and $|V|=n$, then

$\mathcal{E}(\mathcal{H})\geq\sqrt{\frac{(s-1)^{2}}{n}Z(\mathcal{H})+n(n-1)|\det(\mathbf{A})|^{\frac{2}{n}}}.$

###### Proof.

Use Lemma 30 and Remark 29 $(2)$. ∎

######  Theorem 33.

Let $\mathcal{H}=(V,E)$ be a hypergraph, then

$\mathcal{E}(\mathcal{H})\geq\sqrt{\frac{n}{n-1}\lambda_{1}^{2}+n(n-1)|\det(\mathbf{A})|^{\frac{2}{n}}}.$

###### Proof.

Use Lemma 30 and Remark 29 $(3)$. ∎

Note that the bounds obtained in Theorems 31 and 32 can also be compared,
following the same cases listed in Remark 7.2.

Finally, the next remark allow to compare the bounds of the last three
theorems with the bounds of the three preceding theorems.

######  Remark 7.3.

Regarding the bounds of Lemmas 25 and 30, we have four cases:

  1. (1)

If $\mathcal{H}$ is a hypergraph that has zero as an eigenvalue, we have that
the bound of Lemma 25 is sharper than the bound of Lemma 30 (consequently the
bounds from Theorems 26, 27 and 28 are better than the respective bounds from
Theorems 31, 32 and 33).

  2. (2)

If $\mathcal{G}$ is a graph with $|\det(\mathbf{A}(\mathcal{G})|\geq 1$ then
the bound from Lemma 30 is better than the boudn from Lemma 25.

Indeed, we can assume that $\mathcal{G}$ has no isolated vertices. Saying that
30 is better than Lemma 25 means that

$\mathcal{E}(\mathcal{H})^{2}\geq\sum_{i=1}^{n}\lambda_{i}^{2}+n(n-1)|\det(\mathbf{A})|^{\frac{2}{n}}\geq
2\sum_{i=1}^{n}\lambda_{i}^{2}.$

And this is true if, and only if,

$n(n-1)|\det(\mathbf{A})|^{\frac{2}{n}}\geq\sum_{i=1}^{n}\lambda_{i}^{2}.$

and in fact

$n(n-1)|\det(\mathbf{A})|^{\frac{2}{n}}\geq n(n-1)=2\frac{n(n-1)}{2}\geq
2m=\sum_{i=1}^{n}\lambda_{i}^{2}.$

  3. (3)

We have that $(2)$ does not hold for hypergraphs in general. For exemple: Let
$\mathcal{H}=(V,E)$ be a hypergraph with vertices
$V=\left\\{1,2,3,4,5,6,7,8\right\\}$ and edges

$E=\left\\{1234,1237,1238,1256,126,1278,3456,3478,5678\right\\},$

we have $|\det(\mathbf{A}(\mathcal{H}))|=252$ and Lemma 25 is sharper than
Lemma 30, because :

$\sum_{i=1}^{n}\lambda_{i}^{2}>253>224>n(n-1)|\det(\mathbf{A}(\mathcal{H}))|^{\frac{2}{n}}$

  4. (4)

Let $\mathcal{H}$ be a hypergraph with rank $r$ and
$|\det(\mathbf{A}(\mathcal{H})|\geq 1.$ If $n\geq(r-1)\Delta^{2}+1$, the the
bound of Lemma 30 is better than Lemma 25.

Indeed, notice that

$\sum_{i=1}^{n}\lambda_{i}^{2}\leq(r-1)n\Delta^{2}$

.

Therefore

$n(n-1)|\det(\mathbf{A})|^{\frac{2}{n}}\geq n(n-1)\geq
n(r-1)\Delta^{2}\geq\sum_{i=1}^{n}\lambda_{i}^{2}.$

##  8\. Conclusion

In this paper, we made contributions to spectral hypergraph theory. More
precisely, we define the energy of a hypergraph as the energy of its adjacency
matrix, and study its properties. We obtain which hypergraphs have the highest
and lowest energy within the class of hyperstars. By obtaining spectral
properties of operations sum and product of hypergraphs, we prove that the
energy of a hypergraph is never an odd number. We study how the hypergraph
energy varies when we remove a vertex or an edge from it, and we further
define an edge division operation and analyze how an edge division impacts the
energy of a hypergraph. Our main results are the determination of bounds for
the energy. These bounds are functions of well known parameters, such as
maximum and minimum degree, Zagreb index and spectral radius.

We end this paper presenting some open problems about energies of hypergraphs.

  1. (1)

In [5], the authors define the signless Laplacian matrix for general
hypergraphs, so we believe that many results proven in [4] can be generalized
to non-uniform hypergraphs using some of the techniques that we used here.

  2. (2)

We have determined which hyperstar with $t$ vertices have highest and lowest
energy. An interesting problem is to determine in other classes, which is the
hypergraph with the highest and lowest energy.

## Acknowledgments

This article is part of the PhD thesis of Lucas L. Portugal, supervised by
Renata R. Del-Vecchio. Lucas L. Portugal acknowledges the financial support
provided by CAPES and Renata R. Del-Vecchio acknowledges the financial support
by CNPq grant 306262/2019-3. This research took place while Vilmar Trevisan
visited the ‘Dipartimento di Matematica e Applicazioni’, University of Naples
‘Federico II’, Italy and this author acknowledges the financial support
provided by the hosting University, and by CAPES-Print 88887.467572/2019-00,
Brazil, as well as the partial support of CNPq grants 409746/2016-9 and
303334/2016-9, and FAPERGS grant PqG 17/2551-0001.

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SEARCH FOR SUPERSYMMETRY
AT THE LARGE HADRON COLLIDER
T.V. Obikhod
Institute for Nuclear Research, Nat. Acad. of Sci. of Ukraine
47, Prosp. Nauky, Kyiv 03680, Ukraine
V.V. Negliad
Taras Shevchenko National University of Kyiv
6, Prosp. Academician Glushkov, Kyiv 03127, Ukraine
e-mail<EMAIL_ADDRESS>
Abstract

Search for supersymmetry is carried out in the framework of the Minimal
Supersymmetric Standard Model (MSSM). Using the software programs SOFTSUSY and
PROSPINO, the mass spectrum and the production cross-sections of superpartners
are calculated. The results obtained are of importance for searching the new
physics at the LHC.
Keywords: Minimal Supersymmetric Standard Model, mass spectrum, production
cross sections of superpartners.

##  1 Introduction

Problems in high-energy physics associated with unifying all fundamental
interactions in a single theory, the so-called Theory of everything [1], bring
about a necessity of creating high-energy accelerators. On September 10, 2008,
the Large Hadron Collider (LHC) – the largest experimental installation in the
world – was officially started. The LHC can be useful not only in the study of
the Higgs mechanism responsible for the violation of the electroweak
interaction, but also in the verification of the theory of superstrings and
D-branes, because the latter predicts the existence of such objects as

– superpartners,

– Kaluza–Klein particles,

– dark matter candidates, and

– microscopic black holes.

Our work aims at calculating the masses of superpartners at the LHC in the
framework of supersymmetry theory. The latter is one of the most widespread
theories beyond the Standard model (SM), because it allows a number of SM
problems to be solved.

The SM of particle physics is rather a successful theory for the description
of physical phenomena, which can be observed on modern colliders. However, the
theorists are sure that the SM will not work at higher energy scales. Of
course, the SM cannot be an ultimate theory at very high energies, because it
has to be modified in order to include the gravitational interaction on Planck
scales. This problem is called the hierarchy problem [2, 3, 4]. Even if the SM
gives the best description of the subatomic world, it does not provide a
complete picture of the Universe, since it does not imply the unification
mechanism for the strong and electroweak interactions, on the one hand, and
the gravitational interaction, on the other hand [1]. In addition, the
radiation corrections to the mass of a Higgs boson give rise to large
discrepancies between the experiment and the SM theory [5]. A considerable
amount of the cold dark matter and the phenomenon of dark energy, which are
observed in experiment [6], go beyond the scope of SM predictions.

Recent experimental data are bright markers of new physics beyond the SM. In
particular,

(i) new measurements of the Higgs boson properties carried out by ATLAS and
CMS Collaborations [7] point to the physics both in the SM framework and
beyond it;

(ii) there are problems with the stability of electroweak vacuum [8];

(iii) the total width of a recently discovered Higgs boson is 5.4 times larger
than the corresponding SM value [9]; and

(iv) the production cross-sections of $W^{+}W^{-}$ particles obtained on the
LHC in the course of the proton-proton collisions at the energies $\sqrt{s}=7$
and 8 TeV testify to a substantial deviation from the SM [10].

While searching for the supersymmetry, we calculated the production cross-
sections of superpartners at the LHC at the energy
$\sqrt{s}=14\leavevmode\nobreak\ \mathrm{TeV}$ and determined the lower limits
for the superpartner masses.

##  2 Calculations of Mass Spectra and Production
Cross-Sections of Superpartners

Supersymmetry supplementing fermions with bosons and vice versa cannot be an
exact symmetry in the Nature, because fermions and bosons must be degenerate
with respect to the mass. In order to develop a realistic model of high-energy
physics, the supersymmetry has to be broken. The issue concerning a
supersymmetric interaction and the condition of its violation associated with
the inclusion of soft supersymmetry-breaking terms into the interaction
potential is well consistent in the framework of the so-called Minimal
Supersymmetric Standard Model [11].

The MSSM is defined by the superpotential

$W=h^{e}_{ij}L_{i}H_{1}\overline{E}_{j}+h^{d}_{ij}Q_{i}H_{1}\overline{D}_{j}+h^{u}_{ij}Q_{i}H_{2}\overline{U}_{j}+\mu
H_{1}H_{2}$

and the potential of a soft supersymmetry breaking

$V=m_{1}^{2}|H_{1}|^{2}+m_{2}^{2}|H_{2}|^{2}-m_{12}^{2}\left(\\!\epsilon_{ij}H^{i}_{1}H^{j}_{2}+{\rm
h.c.}\\!\right)+$
$+\,M_{\tilde{Q}}^{2}\left[\tilde{t}^{*}_{L}\tilde{t}_{L}+\tilde{b}^{*}_{L}\tilde{b}_{L}\right]+M^{2}_{\tilde{U}}\tilde{t}^{*}_{R}\tilde{t}_{R}\,++\,M^{2}_{\tilde{D}}\tilde{b}^{*}_{R}\tilde{b}_{R}+M^{2}_{\tilde{L}}[\tilde{\nu}^{*}\tilde{\nu}+\tilde{\tau}^{*}_{L}\tilde{\tau}_{L}]\,+$
$+\,M^{2}_{\tilde{E}}\tilde{\tau}^{*}_{R}\tilde{\tau}_{R}+\frac{g}{\sqrt{2}m_{W}}\epsilon_{ij}\biggl{[}\frac{m_{\tau}A_{\tau}}{{\cos}{\beta}}H^{i}_{1}\tilde{l}^{j}_{L}\tau^{*}_{R}\,++\,\frac{m_{b}A_{b}}{{\cos}{\beta}}H^{i}_{1}\tilde{q}^{j}_{L}\tilde{b}^{*}_{R}-\frac{m_{t}A_{t}}{{\sin}{\beta}}H^{i}_{2}\tilde{q}^{j}t^{*}_{R}\biggr{]}+$
$+\frac{1}{2}\biggl{[}M_{3}\overline{\tilde{g}}\tilde{g}+M_{2}\overline{\widetilde{W}^{a}}\widetilde{W}^{a}+M_{1}\overline{\widetilde{B}}\widetilde{B}\biggr{]}\\!,$

where $L_{i}$ and $Q_{i}$ are a slepton and a squark, respectively,
$SU(2)_{L}$ doublets; $\overline{E}_{j}$ and
$(\overline{D}_{j},\overline{U}_{j})$ are a selectron and a squark,
respectively, $SU(2)_{L}$ singlets; and $H_{1}$ and $H_{2}$ are Higgs
$SU(2)_{L}$ doublets.

The superpotential $W$ and the potential $V$ depend on more than 100
parameters. The number of the latter can be reduced to the following five
owing to the theoretical reasoning and the experimental observations [12]:

$m_{0},\leavevmode\nobreak\ m_{1/2},\leavevmode\nobreak\
A_{0},\leavevmode\nobreak\ \tan\beta,\leavevmode\nobreak\
\mathrm{sgn}\leavevmode\nobreak\ \mu,$

where $m_{0}$ and $m_{1/2}$ are the masses of scalar and spinor, respectively,
superpartners; $A_{0}$ is the parameter of the trilinear interaction,
$\tan\beta$ is the ratio between the vacuum expectation values of two Higgs
doublets, and $\mathrm{sgn}\leavevmode\nobreak\ \mu$ is the sign of the Higgs
mixing parameter.

The application of recent experimental data [13] obtained by the ATLAS
Collaboration for the proton-proton collisions with the following final
states: 1) 0 leptons + 2–6 jets, 2) 0 leptons + 7–10 jets, 3) 0 or 1 lepton +
3 b-quark jets, 4) 1 lepton + jets + \+ MET (Missing transverse energy), 5) 1
or 2 tau-leptons + jets + MET, and 6) 2 same-sign leptons + + 0 or more than 3
b-quark jets (see Fig. 1)–allowed us to consider two scenarios of the MSSM
model (they are presented in Table 1)

_Table 1._ Two scenarios of MSSM model

|  $m_{0}$, GeV |  $m_{1/2}$, GeV |  $A_{0}$, GeV | $\mbox{tan}\beta$ | $\mbox{sgn}(\mu)$
---|---|---|---|---|---
I | 1400 | 800 | -2800 | 30 | +1
II | 5000 | 600 | -10000 | 30 | +1

as such that exceed recent experimental observations made in six experiments
shown in Fig. 1.

Figure 1: Gaugino and scalar masses according to the scale of the Grand
Unification Theory

For the calculation of superpartner mass spectra, which are presented in Table
2, we used the software program SOFTSUSY [14].

_Table 2._ Masses of superpartners, GeV

| $m_{\widetilde{u}_{L}}$ | $m_{\widetilde{u}_{R}}$ | $m_{\widetilde{d}_{L}}$ | $m_{\widetilde{d}_{R}}$ | $m_{\widetilde{g}}$ | $m_{\widetilde{\chi}^{0}_{1}}$
---|---|---|---|---|---|---
I | 2188 | 2074 | 2120 | 2069 | 1838 | 344
II | 5079 | 5081 | 5080 | 5079 | 1542 | 273

The presented mass spectrum is confined, because the masses of quark
superpartners are degenerate. In Table 2, the masses of first-generation left-
and right-chiral squarks, (${\widetilde{u}_{L}},{\widetilde{u}_{R}}$) and
(${\widetilde{d}_{L}},{\widetilde{d}_{R}}$), and the masses of gluino,
${\widetilde{g}}$, and neutralino, ${\widetilde{\chi}_{1}^{0}}$(a candidate
for the dark matter), are represented. The center-of-mass energy at the LHC is
planned to achieve a value of 14 TeV, and the luminosity a value of $10^{35}$
$\mathrm{cm}^{-2}\mathrm{s}^{-1}$ in 2015. Therefore, the probability to
observe the processes of superpartner production and subsequent superpartner
decay into quark jets + leptons + MET (e.g., neutralino ${\chi}_{1}^{0}$)
depicted in Fig. 2 will grow considerably.

Using the set of parameters from Table 1, it is possible to calculate the
cross-sections of superpartner production with the help of the software
program PROSPINO [15]. The corresponding results, which are listed in Table 3,
were obtained for the transverse cross-sections with regard for the main terms
$\sigma_{\mathrm{LO}}^{\mathrm{PROSPINO}}$ (LO means “leading order”) for the
squark-squark, squark-gluino, and gluino-gluino production, as well as the
corresponding additional terms $\sigma_{\mathrm{NLO}}^{\mathrm{PROSPINO}}$(NLO
means “next-to-leading order”), which result from making allowance for the
renormalization group terms. The calculations were carried out for the center-
of-mass energy $\sqrt{s}=14\leavevmode\nobreak\ \mathrm{TeV}$.

_Table 3._ LO and NLO transverse cross-sections (in pb units) and
$K$-factors for superpartners

| channel | $\sigma^{\mbox{Prospino}}_{\mbox{LO}}$ | $\sigma^{\mbox{Prospino}}_{\mbox{NLO}}$ | $K^{\mbox{Prospino}}$
---|---|---|---|---
I | squark-squark | 0.446E-02 | 0.488E-02 | 1.096
| squark-gluino | 0.475E-02 | 0.810E-02 | 1.704
| gluino-gluino | 0.631E-03 | 0.204E-02 | 3.226
II | squark-squark | 0.374E-08 | 0.382E-08 | 1.022
| squark-gluino | 0.169 -04 | 0.481 -04 | 2.85
| gluino-gluino | 0.595E-02 | 0.137 -01 | 2.307

The $K$-factor is the ratio between the NLO and LO transverse cross-sections,
$K=\sigma_{\mathrm{NLO}}/\sigma_{\mathrm{LO}}$. While comparing the results of
calculations for two scenarios, we can mark a substantial excess of the cross-
sections in scenario I. This fact is associated with a large difference
between the corresponding values of parameters $m_{0}$ and $A_{0}$ in both
scenarios (see Table 1). Attention should also be paid to a large cross-
section of gluino-gluino production in scenario II, since the mass of such a
gluino amounts to 1542 GeV (Table 2), and the probability of its production
considerably grows with respect to the gluino mass in scenario I. A large
value of $K$-factor testifies to the necessity of more detailed calculations
for the cross-sections, which would include additional (NLO) terms as LO ones
in accordance with the QCD theories.

Figure 2: Processes of superpartner production at the LHC: squark-gluino (left
panel), squark-squark (right upper panel), and gluino-gluino (right bottom
panel) processes.

##  3 Conclusions

Search for the supersymmetry is an important step toward elucidating the deep
contradictions of not only the theoretical, but also experimental character.
This statement is testified by a number of recent experimental data, which are
difficult to be explained if disregarding the new physics, including the
supersymmetry. With the help of the computer simulation (SOFTSUSY, PROSPINO),
leading western scientists [16] carry out a wide range of works in order to
properly respond to the recent experimental data concerning the search for
superpartners. The permanent updating of experimental data associated with the
processing of large data arrays in the GRID database brings about the
necessity of a new experimental simulation with the help of computer programs,
which makes our work up-to-date and challenging, because two new scenarios for
the MSSM parameter space are taken into account in accordance with the last
ATLAS experiment. Since the superpartners are not so heavy as, e.g.,
Kaluza–Klein partners or microscopic black holes, the lower mass limit of
which amounts to approximately 5–6 TeV [17], and the probability of the
superpartner production is high enough, which is confirmed by the results of
our calculations for scenario I, we hope for that they will be observed in the
nearest future, when the energy and the luminosity at the LHC will be
sufficient for that.

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