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

Let H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}$\end{document} be a k-uniform hypergraph on n vertices with degree sequence Δ=d1≥⋯≥dn=δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Delta=d_{1} \geq\cdots\geq d_{n}=\delta$\end{document}. In this paper, in terms of degree di\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d_{i}$\end{document}, we give a new upper bound for the Z-spectral radius of the adjacency tensor of H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}$\end{document}. Some examples are given to show the efficiency of the bound.


Introduction
Let A = (a i 1 i 2 ···i m ) be an mth order n-dimensional real square tensor, x be a real n-vector. Then we define the following real n-vector: If there exist a real vector x and a real number λ such that then λ is called an H-eigenvalue of A and x is called an eigenvector of A associated with λ [1,2]. If there exist a real vector x and a real number λ such that then λ is called a Z-eigenvalue of A and x is called an eigenvector of A associated with λ.
You can see more about the eigenvalues of tensors in [3][4][5][6][7]. Let H be a hypergraph with a vertex set V (H) and an edge set E(H) = {e 1 , e 2 , . . . , e t }. If every edge of H contains exactly k distinct vertices, then H is called a k-uniform hypergraph. The degree of a vertex i in H is the number of edges incident with i, denoted by d i . If d i = d for any i ∈ V (H), then the hypergraph H is called a regular hypergraph. Recently, the spectral radii of hypergraphs have been studied in [8,9].
Let {i 1 , . . . , i k } ∈ E(H) mean that there is an edge containing k distinct vertices i 1 , . . . , i k . Then the adjacency tensor A(H) = (a i 1 ···i k ) of a hypergraph H is a kth order n-dimensional tensor with entries: For a k-uniform hypergraph H, let = d 1 ≥ · · · ≥ d n = δ be the degree sequence of the hypergraph H. In 2013, Xie and Chang [8] presented the following upper bound for the largest Z-eigenvalues ρ Z (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.

Preliminaries
Some basic definitions and useful results are listed as follows.

Definition 2.2 Let
A be an m-order and n-dimensional tensor. We define σ (A) the Zspectrum of A by the set of all Z-eigenvalues of A. Assume σ (A) = ∅, then the Z-spectral radius of 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.
Two useful lemmas are given as follows.
Since {A k } is nonnegative, weakly symmetric, then there exists a nonnegative vector it has a convergent subsequence {y t }. Suppose that y = lim k→∞ y t . By A k y m-1 t = ρ Z (A k )y t , we get Ay m-1 = λy. So λ is an eigenvalue of A. Since λ ≤ ρ Z (A), we have ρ Z (A) = λ.

The Z-spectral radius of tensors and hypergraphs
In this section, let r i (A) = n i 2 ,...,i m =1 |a ii 2 ···i m | -|a ii···i |, we give some bounds on the Zspectral radius of tensors and hypergraphs.  Let u α = max{u i 1 · · · u i m : a i 1 ···i m = 0, 1 ≤ i 1 , . . . , i m ≤ n}, then Suppose that u α = u j 1 · · · u j m . Then, from (2), we can get Then, by u m Therefore, . . , i m ≤ n. Then A + T is an irreducible nonnegative tensor for any chosen positive real number . Now we substitute A + T for A, respectively, in the previous case. When → 0, the result follows by the continuity of ρ Z (A + 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). That is to say, our bound in Theorem 3.2 is always better than the bound in (1). We now show the efficiency of the new upper bound in Theorem 3.2 by the following examples. From Table 1, we can find that bound (3) is always better than (1).

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.