The cyclic index of adjacency tensor of generalized power hypergraphs

TL;DR

This paper investigates the cyclic index of adjacency tensors in generalized power hypergraphs, disproves a conjecture about its behavior, and provides conditions and characterizations related to the index.

Contribution

The paper disproves the conjecture that the cyclic index scales linearly in generalized power hypergraphs and offers new conditions and characterizations for when the conjecture holds.

Findings

Counterexample to the conjecture that c(G^{m,s})=s * c(G)

Sufficient conditions identified for the conjecture to hold

An equivalent matrix equation characterization over _m provided

Abstract

Let $G$ be a $t$-uniform hypergraph, and let $c(G)$ denote the cyclic index of the adjacency tensor of $G$. Let $m,s,t$ be positive integers such that $t \ge 2$, $s \ge 2$ and $m=st$. The generalized power $G^{m,s}$ of $G$ is obtained from $G$ by blowing up each vertex into an $s$-set and preserving the adjacency relation. It was conjectured that $c(G^{m,s})=s \cdot c(G)$. In this paper we show that the conjecture is false by giving a counterexample, and give some sufficient conditions for the conjecture holding. Finally we give an equivalent characterization of the equality in the conjecture by using a matrix equation over $\mathbb{Z}_m$.

Equations

$$a_{i_{1}i_{2}\ldots i_{m}}=\left\{\begin{array}[]{ll}\frac{1}{(m-1)!},&\mbox{if~{}}\{v_{i_{1}},v_{i_{2}},\ldots,v_{i_{m}}\}\in E(G);\\ 0,&\mbox{otherwise}.\end{array}\right.$$

$$\text{mbox}Spec(\mathcal{A})=e^{\mathbf{i}\frac{2\pi }{\ell }}\text{mbox}Spec(\mathcal{A}).$$

$$c(G^{m,s})=s\cdot c(G).$$

$$\begin{array}{rl}\mathcal{A}{x}^m& =\sum _{i_1,i_2,\dots ,i_m\in [n]}{a}_{i_1{i}_2\dots i_m}{x}_{i_1}{x}_{i_2}\cdots x_{i_m},\\ (\mathcal{A}{x}^{m-1}{)}_i& =\sum _{i_2,\dots ,i_m\in [n]}{a}_{i{i}_2\dots i_m}{x}_{i_2}\cdots x_{i_m},i\in [n].\end{array}$$

$$\varphi (i_1)+\cdots +\varphi (i_m)\equiv \frac{m}{\ell }\mathrm{mod}m.$$

$$b_{ev}=\{\begin{array}{ll}1,& \text{mbox}ifv\in e\in E(G),\\ 0,& \text{otherwise}.\end{array}$$

$$B_G\varphi =\frac{m}{\ell }1\text{ over }{\mathbb{Z}}_m,$$

$$B_{G}x=\frac{m}{\ell }1\text{ over }{\mathbb{Z}}_m$$

$$B_{G^{m,s}}\Phi =s\cdot B_G\varphi =\frac{st}{\ell }1=\frac{m}{\ell }1\text{ over }{\mathbb{Z}}_m,$$

$$B_{G^{m,s}}\Phi =t1=\frac{m}{s}1\text{ over }{\mathbb{Z}}_m,$$

$$B_{G^{m,s}}\Phi =\frac{m}{s\cdot {\ell }^{\prime }}1=\frac{t}{{\ell }^{\prime }}1\text{ over }{\mathbb{Z}}_m.$$

$$B_G\varphi =B_{G^{m,s}}\Phi =\frac{t}{{\ell }^{\prime }}1\text{ over }{\mathbb{Z}}_m.$$

$$B_G\varphi =\frac{t}{{\ell }^{\prime }}1\text{ over }{\mathbb{Z}}_t,$$

$$E_1$$

$$E_2$$

$$E_3$$

$$E_4$$

$$c(G^{m,s})\ne s\cdot c(G).$$

$$2ka+2kc=\frac{4k}{\ell }\mathrm{mod}4k,$$

$$B_{G^{m,s}}\Phi =\frac{4ks}{2s}=2k\text{ over }{\mathbb{Z}}_{4ks}.$$

$$B_{G^{m,s}}\Phi =B_G\varphi =2k\text{ over }{\mathbb{Z}}_{4ks}.$$

$$\alpha +3\beta =2\mathrm{mod}4s,3\alpha +\beta =2\mathrm{mod}4s.$$

$$\alpha -\beta =0\mathrm{mod}2s,\alpha +\beta =1\mathrm{mod}s,$$

$$B_{G}x=\frac{t}{c(G)}1\text{ over }{\mathbb{Z}}_m$$

$$B_{G^{m,s}}\Phi =\frac{m}{s\cdot c(G)}1=\frac{t}{c(G)}1\text{ over }{\mathbb{Z}}_m.$$

$$\sum _{u\in \mathbf{v}}\Phi (u)=\varphi (v),\text{ for each }v\in V(G).$$

$$B_{G^{m,s}}\Phi =B_G\varphi =\frac{t}{c(G)}1=\frac{m}{s\cdot c(G)}1\text{ over }{\mathbb{Z}}_m.$$

$$B_{G}x=\frac{t}{c(G)}1\text{ over }{\mathbb{Z}}_t$$

$$(P\mathcal{A}Q{)}_{i_1{i}_2\dots i_m}=p_{i_1{i}_1}{a}_{i_1{i}_2\dots i_m}{q}_{i_2{i}_2}\dots q_{i_m{i}_m}.$$

$$\mathcal{A}=e^{-\mathbf{i}\frac{2\pi }{k}}{D}^{-(m-1)}\mathcal{A}D,$$

$$\text{mbox}Spec(\mathcal{A})=e^{\mathbf{i}\frac{2\pi }{\ell }}\text{mbox}Spec(\mathcal{A}).$$

$$\rho (G^{m,s})e^{\mathbf{i}\frac{2\pi i}{c(G)}}{e}^{\mathbf{i}\frac{2\pi j}{s}},i\in [c(G)],j\in [s].$$

Full text

The first author is the corresponding author, and was supported by National Natural Science Foundation of China #11871073.

1. Introduction

A hypergraph $G=(V(G),E(G))$ consists of a set of vertices, say $V(G)=\{v_{1},v_{2},\cdots v_{n}\}$, and a set of edges, say $E(G)=\{e_{1},e_{2},\cdots e_{k}\}$, where $e_{j}\subseteq V(G)$ for $j\in[k]:=\{1,2,\ldots,k\}$. If $|e_{j}|=m$ for each $j\in[k]$, then $G$ is called an $m$-uniform hypergraph. A walk $W$ in $G$ is a sequence of alternating vertices and edges: $v_{i_{0}},e_{i_{1}},v_{i_{1}},e_{i_{2}},\ldots,e_{i_{l}},v_{i_{l}}$, where $\{v_{i_{j}},v_{i_{j+1}}\}\subseteq e_{i_{j+1}}$ for $j=0,1,\ldots,l-1$. The hypergraph $G$ is connected if every two vertices of $G$ are connected by a walk. The adjacency tensor $\mathcal{A}(G)$ of the hypergraph $G$ is defined as $\mathcal{A}(G)=(a_{i_{1}i_{2}\ldots i_{k}})$ [4], an $m$-th order $n$-dimensional tensor, where

[TABLE]

In general, A tensor (also called hypermatrix) $\mathcal{A}=(a_{i_{1}i_{2}\ldots i_{m}})$ of order $m$ and dimension $n$ over a field $\mathbb{F}$ refers to a multiarray of entries $a_{i_{1}i_{2}\ldots i_{m}}\in\mathbb{F}$ for all $i_{j}\in[n]$ and $j\in[m]$, which can be viewed to be the coordinates of the classical tensor (as a multilinear function) under an orthonormal basis. If $m=2$, then $\mathcal{A}$ is a square matrix of dimension $n$.

In 2005, independently, Lim [13] and Qi [17] introduced eigenvalues for tensors $\mathcal{A}$. Denote by $\rho(\mathcal{A})$ the spectral radius of $\mathcal{A}$, and by $\mbox{Spec}(\mathcal{A})$ the spectrum of $\mathcal{A}$. If $\mathcal{A}$ is further nonnegative, then by Perron-Frobenius theorem of nonnegative tensors, $\rho(\mathcal{A})$ is an eigenvalue of $\mathcal{A}$. Moreover, if $\mathcal{A}$ is weakly irreducible and has $k$ eigenvalues of $\mathcal{A}$ with modulus $\rho(\mathcal{A})$, then those $k$ eigenvalues are equally distributed on the spectral circle. As for nonnegative matrices, the number $k$ is called the cyclic index of $\mathcal{A}$ [2]. The cyclic index reflects the spectral symmetry of nonnegative weakly irreducible tensors, which was generalized and investigated in the paper [5].

Definition 1.1** ([5]).**

Let $\mathcal{A}$ be an $m$-th order $n$-dimensional tensor, and let $\ell$ be a positive integer. The tensor $\mathcal{A}$ is called spectral $\ell$-symmetric if

[TABLE]

The maximum number $\ell$ such that (1.1) holds is called the cyclic index of $\mathcal{A}$ and denoted by $c(\mathcal{A})$, and $\mathcal{A}$ is called spectral $c(\mathcal{A})$-cyclic.

When we say a hypergraph is spectral $\ell$-symmetric or spectral $\ell$-cyclic, this is always referring to its adjacency tensor. In particular, for a uniform hypergraph $G$, denote $c(G):=c(\mathcal{A}(G))$, called the cyclic index of $G$.

For a general tensor $\mathcal{A}$, if it is spectral $\ell$-symmetric, then $\ell|c(\mathcal{A})$ by [5, Lemma 2.7]. It was also proved that if a connected $m$-uniform hypergraph is spectral $\ell$-symmetric, then $\ell|m$, and hence $c(G)|m$; see [5, Lemma 3.2, Corollary 4.3], [6, Lemma 2.5] or [21, Theorem 2.15]. In the paper [5] the authors use the construction of generalized power hypergraphs to show that for every positive integer $m$ and any positive integer $\ell$ such that $\ell|m$, there always exists an $m$-uniform hypergraph $G$ such that $G$ is spectral $\ell$-symmetric. They posed the following conjecture.

Conjecture 1.2** ([5]).**

let $G$ be a $t$-uniform hypergraph, and let $G^{m,s}$ be the generalized power of $G$, where $m=st$. Then

[TABLE]

The generalized power of a hypergraph is defined as follows.

Definition 1.3** ([10]).**

Let $G=(V,E)$ be a $t$-uniform hypergraph. For any integers $m,s$ such that $m>t$ and $1\leq s\leq\frac{m}{t}$, the generalized power of $G$, denoted by $G^{m,s}$, is defined as the $m$-uniform hypergraph with the vertex set $(\cup_{v\in V}\mathbf{v})\cup(\cup_{e\in E}\mathbf{e})$, and the edge set $\{\mathbf{u}_{1}\cup\cdots\cup\mathbf{u}_{t}\cup\mathbf{e}:e=\{u_{1},\ldots,u_{t}\}\in E(G)\}$, where $\mathbf{v}$ denotes an $s$-set corresponding to $v$ and $\mathbf{e}$ denotes an $(m-ts)$-set corresponding to $e$, and all those sets are pairwise disjoint.

In this paper, we only consider the power hypergraphs $G^{m,s}$ with $m=st$, i.e. $G^{m,s}$ is obtained from $G$ by blowing up each vertex into an $s$-set and preserving the adjacency relation. The generalized power hypergraphs include some special cases, such as the powers of simple graphs introduced by Hu, Qi and Shao [9], the generalized powers of simple graphs introduced by Khan and Fan [11]. Peng [16] introduced $s$-paths and $s$-cycles with uniformity $m$ on discussing the Ramsey number, which are exactly the generalized pows of paths and cycles (as simple graphs) respectively if $1\leq s\leq\frac{m}{2}$. The spectral results on generalized power hypergraphs can be found in [9, 24, 11, 23, 12, 10].

For the conjecture 1.2, it was shown that it is true if $c(G)=1$ [5]. In this paper we show that the conjecture is false by giving a counterexample, and give some sufficient conditions for the conjecture holding. We finally give an equivalent characterization of Eq. (1.2) by using a matrix equation over $\mathbb{Z}_{m}$.

2. Preliminaries

2.1. Notions

Let $\mathcal{A}$ be a real tensor of order $m$ and dimension $n$. The tensor $\mathcal{A}$ is called symmetric if its entries are invariant under any permutation of their indices. So, the adjacency tensor of a uniform hypergraph is symmetric.

Given a vector $x\in\mathbb{C}^{n}$, $\mathcal{A}x^{m}\in\mathbb{C}$ and $\mathcal{A}x^{m-1}\in\mathbb{C}^{n}$, which are defined as follows:

[TABLE]

Let $\mathcal{I}=(i_{i_{1}i_{2}\ldots i_{m}})$ be the identity tensor of order $m$ and dimension $n$, that is, $i_{i_{1}i_{2}\ldots i_{m}}=1$ if and only if $i_{1}=i_{2}=\cdots=i_{m}\in[n]$ and $i_{i_{1}i_{2}\ldots i_{m}}=0$ otherwise.

Definition 2.1** ([13, 17]).**

Let $\mathcal{A}$ be an $m$-th order $n$-dimensional real tensor. For some $\lambda\in\mathbb{C}$, if the polynomial system $(\lambda\mathcal{I}-\mathcal{A})x^{m-1}=0$, or equivalently $\mathcal{A}x^{m-1}=\lambda x^{[m-1]}$, has a solution $x\in\mathbb{C}^{n}\backslash\{0\}$, then $\lambda$ is called an eigenvalue of $\mathcal{A}$ and $x$ is an eigenvector of $\mathcal{A}$ associated with $\lambda$, where $x^{[m-1]}:=(x_{1}^{m-1},x_{2}^{m-1},\ldots,x_{n}^{m-1})$.

The determinant of $\mathcal{A}$, denoted by $\det\mathcal{A}$, is defined as the resultant of the polynomials $\mathcal{A}\mathbf{x}^{m-1}$ [8], and the characteristic polynomial $\varphi_{\mathcal{A}}(\lambda)$ of $\mathcal{A}$ is defined as $\det(\lambda\mathcal{I}-\mathcal{A})$ [17, 3]. It is known that $\lambda$ is an eigenvalue of $\mathcal{A}$ if and only if it is a root of $\varphi_{\mathcal{A}}(\lambda)$. The spectrum of $\mathcal{A}$ is the multi-set of the roots of $\varphi_{\mathcal{A}}(\lambda)$.

The spectral symmetry of a connected hypergraph is closed related to a certain coloring of the hypergraph.

Definition 2.2** ([5]).**

Let $m\geq 2$ and $\ell\geq 2$ be integers such that $\ell\mid m$. An $m$-uniform hypergraph $G$ on $n$ vertices is called $(m,\ell)$-colorable if there exists a map $\phi:[n]\to[m]$ such that if $\{i_{1},\ldots,i_{m}\}\in E(G)$, then

[TABLE]

Such $\phi$ is called an $(m,\ell)$-coloring of $G$.

If $m$ is even, an $m$-uniform hypergraph with an $(m,2)$-coloring was called odd-colorable by Nikiforov [14].

Theorem 2.3**.**

[5]** Let $G$ be a connected $m$-uniform hypergraph. Then $G$ is spectral $\ell$-symmetric if and only if $G$ is $(m,\ell)$-colorable.

The edge-vertex incidence matrix $B_{G}=(b_{ev})$ of an $m$-uniform hypergraph $G$ is defined by

[TABLE]

We may view $B_{G}$ as one over $\mathbb{Z}_{m}$, where $\mathbb{Z}_{m}$ is the ring of integers modulo $m$. Now Eq. (2.1) is equivalent to

[TABLE]

where $\phi=(\phi(1),\ldots,\phi(n))$ is considered as a column vector, and $\mathbf{1}$ is an all-ones vector of dimension $n$. So, Theorem 2.3 can be rewritten as follows.

Corollary 2.4**.**

Let $G$ be a connected $m$-uniform hypergraph. Then $G$ is spectral $\ell$-symmetric if and only if the equation

[TABLE]

has a solution.

In Corollary 2.4 and other places of the paper, the number of coordinates of $\mathbf{1}$ is implicated from context, which is equal to the number of vertices of the hypergraph under discussion.

3. Cyclic index of generalized power hypergraphs

Let $G$ be a $t$-uniform hypergraph, and let $G^{m,s}$ be a generalized power hypergraph of $G$, where $1\leq s\leq\frac{m}{t}$. If $m>st$, then each edge of $G$ contains a vertex of degree $1$, and hence $G$ is a $1$-hm bipartite hypergraph [19]. By [19, Theorem 3.2] or [5, Theorem 4.5], $c(G^{m,s})=m$.

So, in the following, we always assume that $G$ is a connected $t$-uniform hypergraph, $m=st$, namely, $G^{m,s}$ is considered to be obtained from $G$ by blowing each vertex $v$ into an $s$-set $\mathbf{v}$ and preserving the adjacency relation. We also assume that the vertex $v$ is contained in $\mathbf{v}$ for each $v\in V(G)$.

Lemma 3.1**.**

If $G$ is spectral $\ell$-symmetric, then $G^{m,s}$ is also spectral $\ell$-symmetric. In particular, $G^{m,s}$ is spectral $c(G)$-symmetric and hence $c(G)|c(G^{m,s})$.

Proof.

Suppose that $G$ is spectral $\ell$-symmetric. By Corollary 2.4, the equation $B_{G}x=\frac{t}{\ell}\mathbf{1}$ has a solution $\phi$ over $\mathbb{Z}_{t}$. Now define a map $\Phi$ on $G^{m,s}$ such that $\Phi|_{\mathbf{v}}=\phi(v)$ for each vertex $v\in V(G)$. Then

[TABLE]

which implies that $G^{m,s}$ is spectral $\ell$-symmetric also by Corollary 2.4. ∎

Lemma 3.2**.**

$G^{m,s}$* is spectral $s$-symmetric.*

Proof.

For each vertex $v\in V(G)$, $v$ is blowing into an $s$-set $\mathbf{v}$ of vertices of $G^{m,s}$, and is assumed to be contained in $\mathbf{v}$. Define a map $\Phi$ on $G^{m,s}$ such that $\Phi(v)=1$ and $\Phi|_{\mathbf{v}\backslash\{v\}}=0$ for each vertex $v\in V(G)$. Then

[TABLE]

which implies that $G^{m,s}$ is spectral $s$-symmetric by Corollary 2.4. ∎

Lemma 3.3**.**

If $G^{m,s}$ is spectral $s\cdot\ell^{\prime}$-symmetric, then $G$ is spectral $\ell^{\prime}$-symmetric.

Proof.

By Corollary 2.4, there exists a map $\Phi$ defined on $G^{m,s}$ such that

[TABLE]

Now define a map $\phi$ on $G$ such that $\phi(v)=\sum_{u\in\mathbf{v}}\Phi(u)$ for each $v\in V(G)$. So we have

[TABLE]

As $m$ is a multiple of $t$,

[TABLE]

which implies that $G$ is spectral $\ell^{\prime}$-symmetric by Corollary 2.4. ∎

By Lemma 3.2, we may assume $c(G^{m,s})=s\cdot\ell^{\prime}$, where $\ell^{\prime}$ is a positive integer. By Lemma 3.3, we know that $G$ is spectral $\ell^{\prime}$-symmetric and hence $\ell^{\prime}|c(G)$ by [5, Lemma 2.7]. So we get the following result immediately.

Corollary 3.4**.**

$c(G^{m,s})|s\cdot c(G)$.

Corollary 3.5**.**

$G^{m,s}$ is spectral $\frac{s\cdot c(G)}{(s,c(G))}$-symmetric. In particular, if $(s,c(G))=1$ or $(s,t)=1$, then $c(G^{m,s})=s\cdot c(G)$.

Proof.

By Lemma 3.1 and Lemma 3.2, we know that $c(G)|c(G^{m,s})$ and $s|c(G^{m,s})$, implying that $\frac{s\cdot c(G)}{(s,c(G))}|c(G^{m,s})$. So, $G^{m,s}$ is spectral $\frac{s\cdot c(G)}{(s,c(G))}$-symmetric. As $c(G)|t$, if $(s,t)=1$, then $(s,c(G))=1$. If $(s,c(G))=1$, then $s\cdot c(G)|c(G^{m,s})$. Then result follows by Corollary 3.4. ∎

By Corollary 3.5, Conjecture 1.2 holds in some special cases, including the case of $c(G)=1$. However, Conjecture 1.2 does not hold in general. Now we give a counterexample to show the negative answer to the conjecture.

Definition 3.6** ([14]).**

Let $n\geq 16k$ and let partition $[n]$ into three sets $A,B,C$ such that $|A|\geq 6k$, $|B|\geq 6k$ and $|C|\geq 4k$. Define the four families of $4k$-subsets of $[n]$.

[TABLE]

Now define a $4k$-uniform hypergraph $G$ by setting $V(G)=[n]$ and $E(G)=E_{1}\cup E_{2}\cup E_{3}\cup E_{4}$. We call $G$ a Nikiforov’s hypergraph as it is introduced by Nikiforov.

Nikiforov [14] showed that Nikiforov’s hypergraphs $G$ are odd-colorable, or $(4k,2)$-colorable in terms our definition, by defining a function $\phi$ on $G$ such that $\phi|_{A}=1$, $\phi|_{B}=4k-1$ and $\phi|_{C}=0$. By Theorem 2.3, $G$ is spectral $2$-symmetric.

By the following result, if $G$ is a Nikiforov’s hypergraph and $s$ is even, then

[TABLE]

So we give a negative answer to Conjecture 1.2.

Theorem 3.7**.**

Let $G$ be a $4k$-uniform Nikiforov’s hypergraph. Then the following results hold.

1. (1)

$c(G)=2$. 2. (2)

If $s$ is even, then $c(G^{m,s})=s$.

Proof.

(1) We first show that $c(G)=2$. Suppose that $G$ is spectral $\ell$-symmetric. Then there exists a $\phi:[n]\to[4k]$ such that $B_{G}\phi=\frac{4k}{\ell}$ over $\mathbb{Z}_{4k}$. It is easily seen that $\phi$ is constant on each of $A,B,C$ by the equation. So, let $\phi|_{A}:=a$, $\phi|_{B}:=b$ and $\phi|_{C}:=c$. Then, by considering the edges in $E_{1}$, we have

[TABLE]

which implies that $\ell$ equals $1$ or $2$, and hence $c(G)=2$ as $G$ is spectral $2$-symmetric.

(2) By Corollary 3.4, $c(G^{m,s})|2s$, where $m=4ks$. By Lemma 3.2, $G^{m,s}$ is spectral $s$-symmetric, and hence $s|c(G^{m,s})$. We will show that if $s$ is even, then $G^{m,s}$ is not spectral $2s$-symmetric so that $c(G^{m,s})=s$.

Assume to the contrary that $G^{m,s}$ is spectral $2s$-symmetric. Then there exists a $\Phi:V(G^{m,s})\to[4ks]$ such that

[TABLE]

For each $v\in V(G)$, define $\phi(v):=\sum_{u\in\mathbf{v}}\Phi(u)$. So we have

[TABLE]

It is also easily seen that $\phi|_{A}:=\alpha$, $\phi|_{B}:=\beta$ and $\phi|_{C}:=\iota$. By considering the edges in $E_{3}$ and $E_{4}$ respectively, we have

[TABLE]

So

[TABLE]

which yields a contradiction as $s$ is an even number. ∎

Finally we give an equivalent characterization of Eq. (1.2) in Conjecture 1.2.

Theorem 3.8**.**

$c(G^{m,s})=s\cdot c(G)$* if and only if the equation*

[TABLE]

has a solution.

Proof.

Suppose that $c(G^{m,s})=s\cdot c(G)$. Then $G^{m,s}$ is spectral $s\cdot c(G)$-symmetric, and by Corollary 2.4, there exists a map $\Phi:V(G^{m,s})\to[m]$ such that

[TABLE]

For each $v\in V(G)$, define $\phi(v):=\sum_{u\in\mathbf{v}}\Phi(u)$. So we have $B_{G^{m,s}}\Phi=B_{G}\phi$, and get the necessity.

On the other hand, if $B_{G}x=\frac{t}{c(G)}\mathbf{1}$ has a solution $\phi$ over $\mathbb{Z}_{m}$. Define a map $\Psi:V(G^{m,s})\to[m]$ such that

[TABLE]

There are $|V(G)|$ independent linear equations; such $\Phi$ is easily got (e.g. for each $v\in V(G)$, take $\Phi(v)=\phi(v)$ and $\Phi(u)=0$ for each $u\in\mathbf{v}\backslash\{v\}$). So we have

[TABLE]

So $G^{m,s}$ is spectral $s\cdot c(G)$-symmetric. The sufficiency follows by Corollary 3.4. ∎

As $G$ is spectral $c(G)$-symmetric, by Corollary 2.4 the equation

[TABLE]

has a solution. Obviously, if the equation (3.1) has a solution, then the equation (3.2) has a solution as $m$ is a multiple of $t$. However, the converse does not hold in general; see the previous counterexample.

4. Remark

For a nonnegative weakly irreducible tensor $\mathcal{A}$, its cyclic index $c(\mathcal{A})$ is exactly the number of eigenvalues with modulus $\rho(\mathcal{A})$. The is implied by Perron-Frobenius theorem for nonnegative tensors, where an eigenvalue of $\mathcal{A}$ is called $H^{+}$-eigenvalue (respectively $H^{++}$-eigenvalue) if it is associated with a nonnegative (respectively positive) eigenvector. For the notion of irreducible or weakly irreducible tensors, one can refer to [1] and [7]. It is known that the adjacency tensor of a uniform hypergraph $G$ is weakly irreducible if and only if $G$ is connected [15, 22].

Theorem 4.1** (The Perron-Frobenius Theorem for nonnegative tensors).**

1. (1)

(Yang and Yang [22])* If $\mathcal{A}$ is a nonnegative tensor, then $\rho(\mathcal{A})$ is an $H^{+}$-eigenvalue of $\mathcal{A}$.* 2. (2)

(Friedland, Gaubert and Han [7])* If furthermore $\mathcal{A}$ is weakly irreducible, then $\rho(\mathcal{A})$ is the unique $H^{++}$-eigenvalue of $\mathcal{A}$, with a unique positive eigenvector, up to a positive scalar.* 3. (3)

(Chang, Pearson and Zhang [1])* If moreover $\mathcal{A}$ is irreducible, then $\rho(\mathcal{A})$ is the unique $H^{+}$-eigenvalue of $\mathcal{A}$, with a unique nonnegative eigenvector, up to a positive scalar.*

According to the definition of tensor product in [18], for a tensor $\mathcal{A}$ of order $m$ and dimension $n$, and two diagonal matrices $P,Q$ both of dimension $n$, the product $P\mathcal{A}Q$ has the same order and dimension as $\mathcal{A}$, whose entries are defined by

[TABLE]

If $P=Q^{-1}$, then $\mathcal{A}$ and $P^{m-1}\mathcal{A}Q$ are called diagonal similar. It is proved that two diagonal similar tensors have the same spectrum [18].

Theorem 4.2** ([22]).**

Let $\mathcal{A}$ and $\mathcal{B}$ be two $m$-th order $n$-dimensional real tensors with $|\mathcal{B}|\leq\mathcal{A}$, namely, $|b_{i_{1}i_{2}\ldots i_{m}}|\leq a_{i_{1}i_{2}\ldots i_{m}}$ for each $i_{j}\in[n]$ and $j\in[m]$. Then

1. (1)

$\rho(\mathcal{B})\leq\rho(\mathcal{A})$. 2. (2)

Furthermore, if $\mathcal{A}$ is weakly irreducible and $\rho(\mathcal{B})=\rho(\mathcal{A})$, where $\lambda=\rho(\mathcal{A})e^{\mathbf{i}\theta}$ is an eigenvalue of $\mathcal{B}$ corresponding to an eigenvector $y$, then $y$ contains no zero entries, and $\mathcal{B}=e^{-\mathbf{i}\theta}D^{-(m-1)}\mathcal{A}D$, where $D=\hbox{diag}\{\frac{y_{1}}{|y_{1}|},\ldots,\frac{y_{n}}{|y_{n}|}\}$.

Theorem 4.3** ([22]).**

Let $\mathcal{A}$ be an $m$-th order $n$-dimensional weakly irreducible nonnegative tensor. Suppose $\mathcal{A}$ has $k$ distinct eigenvalues with modulus $\rho(\mathcal{A})$ in total. Then these eigenvalues are $\rho(\mathcal{A})e^{\mathbf{i}\frac{2\pi j}{k}}$, $j=0,1,\ldots,k-1$. Furthermore,

[TABLE]

and the spectrum of $\mathcal{A}$ remains invariant under a rotation of angle $\frac{2\pi}{k}$ (but not a smaller positive angle) of the complex plane.

Suppose $\mathcal{A}$ be as in Theorem 4.3. If $\mbox{Spec}(\mathcal{A})$ is invariant under a rotation of angle $\theta$ of the complex plane, i.e. $\mbox{Spec}(\mathcal{A})=e^{\mathbf{i}\theta}\mbox{Spec}(\mathcal{A})$, then $\rho(\mathcal{A})e^{\mathbf{i}\theta}$ is an eigenvalue of $\mathcal{A}$ by Theorem 4.1. By Theorem 4.3, $\theta=\frac{2\pi j}{k}$ for some $j\in[k]$, and hence by Theorem 4.2 (and taking $\mathcal{B}=\mathcal{A}$), $\mbox{Spec}(\mathcal{A})=e^{\mathbf{i}\frac{2\pi j}{k}}\mbox{Spec}(\mathcal{A})$. So, for some positive integer $\ell$, $\ell|k$,

[TABLE]

The number $k$ in Theorem 4.3 is exactly the cyclic index of $\mathcal{A}$. In addition, if $\mathcal{A}$ is spectral $\ell$-symmetric, Then $\ell\mid c(\mathcal{A})$ by Theorem 4.3.

Now return to a connected $t$-uniform hypergraph $G$ and its power $G^{m,s}$, where $m=st$. By Lemma 3.1, $G^{m,s}$ is spectral $c(G)$-symmetric; and by Lemma 3.2, $G^{m,s}$ is also spectral $s$-symmetric. So $G^{m,s}$ has eigenvalues

[TABLE]

In particular, $\rho(G^{m,s})e^{\mathbf{i}\frac{2\pi}{d}}$ is an eigenvalue of $G^{m,s}$, where $d=\frac{s\cdot c(G)}{(s,c(G))}$. So by Theorem 4.2, $G^{m,s}$ is spectral $d$-symmetric, which is consistent with Corollary 3.5.