The trace and Estrada index of uniform hypergraphs with cut vertices

TL;DR

This paper derives formulas for the traces and Estrada index of uniform hypergraphs with cut vertices, revealing extremal properties of hyperpaths and hyperstars among hypertrees.

Contribution

It provides explicit trace formulas for hypergraphs with cut vertices and characterizes extremal hypergraphs for Estrada index among hypertrees.

Findings

Hyperpath has minimum Estrada index among hypertrees with fixed edges.

Hyperstar has maximum Estrada index among hypertrees with fixed edges.

Formulas for traces of adjacency tensors in hypergraphs with cut vertices.

Abstract

Let $\mathcal{H}$ be an $m$-uniform hypergraph, and let $\mathcal{A}(\mathcal{H})$ be the adjacency tensor of $\mathcal{H}$ which can be viewed as a system of homogeneous polynomials of degree $m-1$. Morozov and Shakirov generalized the traces of linear systems to nonlinear homogeneous polynomial systems and obtained explicit formulas for multidimensional resultants. Sun, Zhou and Bu introduced the Estrada index of uniform hypergraphs which is closely related to the traces of their adjacency tensors. In this paper we give formulas for the traces of $\mathcal{A}(\mathcal{H})$ when $\mathcal{H}$ contains cut vertices, and obtain results on the traces and Estrada index when $\mathcal{H}$ is perturbed under local changes. We prove that among all hypertrees with fixed number of edges, the hyperpath is the unique one with minimum Estrada index and the hyperstar is the unique one with maximum Estrada index.