Inverse Perron Values and Connectivity of a Uniform Hypergraph

Changjiang Bu,Haifeng Li,Jiang Zhou

doi:10.37236/7410

Abstract

In this paper, we show that a uniform hypergraph $\mathcal{G}$ is connected if and only if one of its inverse Perron values is larger than $0$. We give some bounds on the bipartition width, isoperimetric number and eccentricities of $\mathcal{G}$ in terms of inverse Perron values. By using the inverse Perron values, we give an estimation of the edge connectivity of a $2$-design, and determine the explicit edge connectivity of a symmetric design. Moreover, relations between the inverse Perron values and resistance distance of a connected graph are presented.

Highlights

Let V (G) and E(G) denote the vertex set and edge set of a hypergraph G, respectively
Let R[m,n] denote the set of order m dimension n real tensors, and let Rn+ denote the cone of nonnegative vectors in Rn
[2] Let G be a connected graph with n vertices and i ∈ [n]

Summary

Introduction

Let V (G) and E(G) denote the vertex set and edge set of a hypergraph G, respectively. For any vertex j of a k-uniform hypergraph G, we define the inverse Perron value of j as n αj(G) = min LGxk : x ∈ Rn+, xki = 1, xj = 0. J in a connected graph G, the resistance distance between i and j, denoted by rij(G), is defined to be the effective resistance between them when unit resistors are placed on every edge of G. We use the inverse Perron values to estimate the edge connectivity of 2-designs.

Preliminaries

Inverse Perron values of uniform hypergraphs

Inverse Perron values and resistance distance of graphs