# An upper bound for the *Z*-spectral radius of adjacency tensors

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## Abstract

Let \(\mathcal{H}\) be a *k*-uniform hypergraph on *n* vertices with degree sequence \(\Delta=d_{1} \geq\cdots\geq d_{n}=\delta\). In this paper, in terms of degree \(d_{i}\), we give a new upper bound for the *Z*-spectral radius of the adjacency tensor of \(\mathcal{H}\). Some examples are given to show the efficiency of the bound.

## Keywords

Hypergraph*Z*-eigenvalue Bound Nonnegative tensor

## MSC

15A18 15A69 65F15 65F10## 1 Introduction

*m*th order

*n*-dimensional real square tensor,

*x*be a real

*n*-vector. Then we define the following real

*n*-vector:

*x*and a real number

*λ*such that

*λ*is called an H-eigenvalue of \(\mathcal{A}\) and

*x*is called an eigenvector of \(\mathcal{A}\) associated with

*λ*[1, 2]. If there exist a real vector

*x*and a real number

*λ*such that

*λ*is called a

*Z*-eigenvalue of \(\mathcal{A}\) and

*x*is called an eigenvector of \(\mathcal{A}\) associated with

*λ*. You can see more about the eigenvalues of tensors in [3, 4, 5, 6, 7].

Let \(\mathcal{H}\) be a hypergraph with a vertex set \(V (\mathcal{H}) \) and an edge set \(E(\mathcal{H})= \{e_{1},e_{2},\ldots,e_{t}\}\). If every edge of \(\mathcal{H}\) contains exactly *k* distinct vertices, then \(\mathcal{H}\) is called a *k*-uniform hypergraph. The degree of a vertex *i* in \(\mathcal{H}\) is the number of edges incident with *i*, denoted by \(d_{i}\). If \(d_{i} = d\) for any \(i \in V (\mathcal {H})\), then the hypergraph \(\mathcal{H}\) is called a regular hypergraph. Recently, the spectral radii of hypergraphs have been studied in [8, 9].

*k*distinct vertices \(i_{1}, \ldots, i_{k}\). Then the adjacency tensor \(\mathcal{A}(\mathcal{H})=(a_{i_{1}\cdots i_{k}})\) of a hypergraph \(\mathcal{H}\) is a

*k*th order

*n*-dimensional tensor with entries:

*Z*-eigenvalues of the adjacency tensor \(\mathcal{A}(\mathcal{H})\) is denoted by \(\rho_{Z}(\mathcal{H})\), which is called the

*Z*-spectral radius of the adjacency tensor \(\mathcal{A}(\mathcal{H})\).

*k*-uniform hypergraph \(\mathcal{H}\), let \(\Delta=d_{1} \geq\cdots \geq d_{n}=\delta\) be the degree sequence of the hypergraph \(\mathcal{H}\). In 2013, Xie and Chang [8] presented the following upper bound for the largest

*Z*-eigenvalues \(\rho_{Z} (\mathcal {H})\) of adjacency tensors:

In this paper, we give a new upper bounds in terms of degree \(d_{i}\) for the *Z*-spectral radius of hypergraphs, which improves the bound as shown in (1). Then we give some examples to compare these bounds for *Z*-spectral radius of hypergraphs.

## 2 Preliminaries

Some basic definitions and useful results are listed as follows.

### Definition 2.1

([10])

The tensor \(\mathcal {A}\) is called reducible if there exists a nonempty proper index subset \(\mathbb{J} \subset\{ 1,2,\ldots ,n \}\) such that \(a_{i_{1},i_{2},\ldots,i_{m}} = 0\), \(\forall i_{1} \in\mathbb{J}\), \(\forall i_{2}, \ldots, i_{m} \notin\mathbb{J}\). If \(\mathcal{A}\) is not reducible, then we call \(\mathcal{A}\) to be irreducible.

### Definition 2.2

*Z*-spectrum of \(\mathcal{A}\) by the set of all

*Z*-eigenvalues of \(\mathcal{A}\). Assume \(\sigma(\mathcal{A})\neq \emptyset\), then the

*Z*-spectral radius of \(\mathcal{A}\) is denoted by

The concept of *weakly symmetric* was first introduced and used by Chang, Pearson, and Zhang [11] in order to study the following Perron–Frobenius theorem for the *Z*-eigenvalue of nonnegative tensors.

### Lemma 2.1

([11])

*Let*\(\mathcal{A}=(a_{i_{1}i_{2} \cdots i_{m}})\)*be a weakly symmetric nonnegative tensor*, *then the spectral radius*\(\rho _{Z} (\mathcal{A})\)*is a positive**Z*-*eigenvalue with a nonnegative**Z*-*eigenvector**x*. *Furthermore*, *if*\(\mathcal{A}\)*is irreducible*, *x**is positive*.

\(|\mathcal{A}|\) means that \((|\mathcal{A}|)_{i_{1} \cdots i_{m}}=|a_{i_{1} \cdots i_{m}}|\). Two useful lemmas are given as follows.

### Lemma 2.2

*Let*\(\mathcal{A}\)*and*\(\mathcal{B}\)*be two weakly symmetric and irreducible tensors of order**m**and dimension**n*. *If*\(\mathcal{B}\)*and*\(\mathcal{B} - |\mathcal{A}|\)*are nonnegative*, *then*\(\rho_{Z}(\mathcal{B}) \geq\rho_{Z}(|\mathcal{A}|)\).

### Proof

*y*be the eigenvector associated with

*β*, where

*β*is a

*Z*-eigenvalue of \(\mathcal{A}\). Then we can get

### Lemma 2.3

*Let*\(\{\mathcal{A}_{k}\}\)

*be a sequence of nonnegative*,

*weakly symmetric tensors of order*

*m*

*and dimension*

*n*,

*and*\(\mathcal{A}_{k} - \mathcal{A}_{k+1}\)

*be nonnegative for each positive integer*

*k*.

*Then*

### Proof

*λ*is an eigenvalue of \(\mathcal{A}\). Since \(\lambda\leq\rho_{Z}(\mathcal{A})\), we have \(\rho _{Z}(\mathcal{A}) = \lambda\). □

## 3 The *Z*-spectral radius of tensors and hypergraphs

In this section, let \(r_{i}(\mathcal{A})=\sum_{i_{2},\ldots,i_{m} = 1}^{n} |a_{ii_{2}\cdots i_{m}}|-|a_{ii\cdots i}|\), we give some bounds on the *Z*-spectral radius of tensors and hypergraphs.

### Theorem 3.1

*Let*\(\mathcal{A}\)

*be weakly symmetric nonnegative tensors of order*

*m*

*and dimension n*.

*Then*

### Proof

*Z*-eigenvalues \(\rho_{Z} (\mathcal{A})\) of \(\mathcal{A}\). Then

Case 2. If \(\mathcal{A}\) is reducible. Let \(\mathcal{T} =(t_{i_{1}i_{2} \cdots i_{m}})\), \(t_{i_{1}i_{2} \cdots i_{m} }=1\) for all \(1\leq i_{1},i_{2}, \ldots , i_{m} \leq n \). Then \(\mathcal{A} + \epsilon\mathcal{T}\) is an irreducible nonnegative tensor for any chosen positive real number *ϵ*. Now we substitute \(\mathcal{A} + \epsilon\mathcal{T}\) for \(\mathcal{A}\), respectively, in the previous case. When \(\epsilon \rightarrow0\), the result follows by the continuity of \(\rho_{Z}(\mathcal {A} + \epsilon\mathcal{T} )\). □

By Theorem 3.1, a bound on the *Z*-spectral radius of a uniform hypergraph is obtained, we also compare the bound with the result in (1).

### Theorem 3.2

*Let*\(\mathcal{H}\)

*be a*

*k*-

*uniform hypergraph on*

*n*

*vertices with the degree sequence*\(\Delta=d_{1} \geq\cdots\geq d_{n}=\delta\).

*Then*

### Proof

Case 2. If \(\mathcal{A}(\mathcal{H})\) is reducible. Let \(\mathcal{T} =(t_{i_{1}i_{2} \cdots i_{k} })\), \(t_{i_{1}i_{2} \cdots i_{k} }=1\), for all \(1\leq i_{1},i_{2}, \ldots, i_{k} \leq n \). Then \(\mathcal{A}(\mathcal{H}) + \epsilon\mathcal{T}\) is an irreducible nonnegative tensor for any chosen positive real number *ϵ*. Now we substitute \(\mathcal {A}(\mathcal{H}) + \epsilon\mathcal{T}\) for \(\mathcal{A}(\mathcal {H})\), respectively, in the previous case. When \(\epsilon\rightarrow 0\), the result follows by the continuity of \(\rho_{Z}(\mathcal{A}(\mathcal {H}) + \epsilon\mathcal{T} )\). □

### Remark

We now show the efficiency of the new upper bound in Theorem 3.2 by the following examples.

### Example 1

Consider 3-uniform hypergraph \(\mathcal{H}_{1}\) with a vertex set \(V(\mathcal{H}_{1})=\{1,2,3,4,5,6,7\}\) and an edge set \(E(\mathcal{H}_{1}) =\{e_{1}, e_{2} , e_{3}\}\), where \(e_{1} = \{1, 2, 3\}\), \(e_{2} = \{1, 4, 5\}\), \(e_{3} = \{1, 6, 7\}\).

### Example 2

Consider 3-uniform hypergraph \(\mathcal{H}_{2}\) with a vertex set \(V(\mathcal{H}_{2})=\{1,2,3,4,5,6,7\}\) and an edge set \(E(\mathcal{H}_{2}) =\{e_{1}, e_{2} , e_{3}\}\), where \(e_{1} = \{1, 6, 7\}\), \(e_{2} = \{2, 6, 7\}\), \(e_{3} = \{3, 6, 7\}\).

## 4 Conclusion

In this paper, we get a new bound for the *Z*-spectral radius of tensors. As applications, in terms of the degree sequence \(d_{i}\), we obtain a new bound for the *Z*-spectral radius of hypergraphs, which is always better than the bound in [8]. We list two examples to show the efficiency of our new bound.

## Notes

### Acknowledgements

Wu is supported by the Research Center for Qianbei Culture of Guizhou Higher Education Humanistic and Social Science Research Base Foundation: Kelaofolk Mathematical Investigation and Cultural Inheritance[2015JD114]. He is supported by the Science and Technology Foundation of Guizhou Province (Qian ke he Ji Chu [2016]1161); Guizhou Province Natural Science Foundation in China (Qian Jiao He KY [2016]255); the Doctoral Scientific Research Foundation of Zunyi Normal College (BS[2015]09); High-level Innovative Talents of Guizhou Province (Zun Ke He Ren Cai [2017]8). Liu is supported by the National Science Foundation of China (71461027); Science and Technology Talent Training Object of Guizhou Province Outstanding Youth (Qian ke he ren zi [2015]06); Guizhou Province Natural Science Foundation in China (Qian Jiao He KY [2014]295); 2013, 2014, and 2015 Zunyi 15851 Talents Elite Project funding; Zhunyi Innovative Talent Team (Zunyi KH (2015)38). Tian is supported by Guizhou Province Natural Science Foundation in China (Qian Jiao He KY [2015]451); Science and Technology Foundation of Guizhou Province (Qian ke he J zi [2015]2147).

### Authors’ contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

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