High-ordered spectral characterization of unicyclic graphs

TL;DR

This paper investigates the spectral properties of unicyclic graphs using tensor traces, establishing conditions under which these graphs are uniquely determined by their high-ordered spectra and providing examples of cospectral pairs.

Contribution

It introduces formulas for traces of unicyclic graphs' powers and proves that certain classes are uniquely determined by their high-ordered spectra.

Findings

Identified formulas for traces of unicyclic graphs' powers.

Proved some unicyclic graphs are determined by high-ordered spectra.

Presented examples of cospectral unicyclic graphs with different high-ordered spectra.

Abstract

In this paper we will apply the tensor and its traces to investigate the spectral characterization of unicyclic graphs. Let $G$ be a graph and $G^m$ be the $m$-th power (hypergraph) of $G$. The spectrum of $G$ is referring to its adjacency matrix, and the spectrum of $G^m$ is referring to its adjacency tensor. The graph $G$ is called determined by high-ordered spectra (DHS for short) if, whenever $H$ is a graph such that $H^m$ is cospectral with $G^m$ for all $m$, then $H$ is isomorphic to $G$. In this paper we first give formulas for the traces of the power of unicyclic graphs, and then provide some high-ordered cospectral invariants of unicyclic graphs. We prove that a class of unicyclic graphs with cospectral mates is DHS, and give two examples of infinitely many pairs of cospectral unicyclic graphs but with different high-ordered spectra.