The trace of uniform hypergraphs with application to Estrada index

TL;DR

This paper studies the traces of hypergraph adjacency tensors, deriving new formulas and applying them to identify hypergraphs with maximum Estrada index under certain constraints.

Contribution

It introduces new trace formulas for hypertrees and unicyclic hypergraphs and applies these to find extremal hypergraphs with maximum Estrada index.

Findings

Identifies the hypertree with maximum Estrada index for fixed edges and matchings.

Determines the unicyclic hypergraph with maximum Estrada index for fixed edges and girth 3.

Provides perturbation results for traces of hypergraphs with cut vertices.

Abstract

In this paper we investigate the traces of the adjacency tensor of hypergraphs (simply called the traces of hypergraphs). We give new expressions for the traces of hypertrees and linear unicyclic hypergraphs by the weight function assigned to their connected sub-hypergraphs, and provide some perturbation results for the traces of a hypergraph with cut vertices. As applications we determine the unique hypertree with maximum Estrada index among all hypertrees with fixed number of edges and perfect matchings, and the unique unicyclic hypergraph with maximum Estrada index among all unicyclic hypergraph with fixed number of edges and girth $3$.