Inverse Perron values and connectivity of a uniform hypergraph

TL;DR

This paper explores the relationship between inverse Perron values and various connectivity measures in uniform hypergraphs, providing bounds, estimations, and explicit calculations for edge connectivity and resistance distances.

Contribution

It introduces the use of inverse Perron values to characterize hypergraph connectivity and derive bounds on key graph invariants, including edge connectivity and resistance distance.

Findings

A uniform hypergraph is connected iff one inverse Perron value is positive.

Bounds on bipartition width, isoperimetric number, and eccentricities are established.

Explicit edge connectivity for symmetric designs is determined.

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.