High-ordered spectral characterizations of graphs

TL;DR

This paper introduces the concept of high-ordered spectra of graphs, using it to distinguish non-isomorphic graphs and prove that Smith's graphs are uniquely determined by this spectrum.

Contribution

It defines high-ordered spectra for graphs, demonstrates their effectiveness in graph isomorphism, and shows that Smith's graphs are uniquely identified by this spectrum.

Findings

Smith's graphs are determined by the high-ordered spectrum.

Infinitely many non-isomorphic trees share the same spectrum but differ in high-ordered spectrum.

High-ordered spectrum can distinguish certain cospectral trees.

Abstract

The spectrum of the $k$-power hypergraph of a graph $G$ is called the $k$-ordered spectrum of $G$.If graphs $G_1$ and $G_2$ have same $k$-ordered spectrum for all positive integer $k\geq2$, $G_1$ and $G_2$ are said to be high-ordered cospectral. If all graphs who are high-ordered cospectral with the graph $G$ are isomorphic to $G$, we say that $G$ is determined by the high-ordered spectrum.In this paper, we use the high-ordered spectrum of graphs to study graph isomorphism and show that all Smith's graphs are determined by the high-ordered spectrum.And we give infinitely many pairs of trees with same spectrum but different high-ordered spectrum by high-ordered cospectral invariants of trees,it means that we can determine that these cospectral trees are not isomorphism by the high-ordered spectrum.