Some Upper Bounds of Maximum <I>E</I>-Eigenvalues of Uniform Hypergraphs

Abstract

A hypergraphs, as a generalization of a general graph, is often used as an effective tool to describe complex structures in discrete mathematics, computer science and other fields. Hypergraph theory and related parameters of hypergraph are important research topics in hypergraph theory. In particular, the problem of spectral extremum of graphs has been widely concerned. This problem originates from the problem proposed by Brualdi and Solheid in 1986. That is to find the upper and lower bounds of spectral radius of a given graph class and characterize the polar graph that reaches the upper and lower bounds. Let <I>H </I>be a uniform hypergraph. Let <I>A</I>(<I>H</I>) be the adjacency tensor of <I>H</I>. In this work, by using Perron-Frobenius theorem, Hölder’s inequality and inequality of arithmetic and geometric means, we establish some upper bounds for the maximum <I>E</I>-eigenvalue of a uniform hypergraph instead of the degrees of vertices and edge number of hypergraph H. In addition, we characterize the extremal hypergraphs that reach the upper bounds.