Constructing the $r$-uniform supertrees with the same spectral radius and matching energyv

TL;DR

This paper introduces the concept of matching energy for r-uniform hypergraphs and constructs supertrees with identical spectral radius and matching energy, revealing new spectral properties and extending known graph spectral results.

Contribution

It defines matching energy for r-uniform hypergraphs and constructs supertrees with identical spectral radius and matching energy, expanding spectral graph theory to hypergraphs.

Findings

Constructed three pairs of supertrees with same spectral radius and matching energy.

Characterized two infinite families of supertrees with identical spectral properties.

Extended known graph spectral results to hypergraph context.

Abstract

An $r$-uniform supertree is a connected and acyclic hypergraph of which each edge has $r$ vertices, where $r\geq 3$. We propose the concept of matching energy for an $r$-uniform hypergraph, which is defined as the sum of the absolute value of all the eigenvalues of its matching polynomial. With the aid of the matching polynomial of an $r$-uniform supertree, three pairs of $r$-uniform supertrees with the same spectral radius and the same matching energy are constructed, and two infinite families of $r$-uniform supertrees with the same spectral radius and the same matching energy are characterized. Some known results about the graphs with the same spectra regarding to their adjacency matrices can be naturally deduced from our new results.