Generalized Spectral Characterization of Graphs: Revisited

TL;DR

This paper reviews and extends methods for identifying graphs uniquely determined by their generalized spectra, enhancing understanding of spectral graph characterization.

Contribution

It revisits previous results and introduces new criteria for a graph to be uniquely identified by its generalized spectrum.

Findings

Reviewed existing spectral characterization methods

Proposed new criteria for DGS graphs

Enhanced understanding of spectral graph uniqueness

Abstract

A graph $G$ is said to be \textit{determined by its generalized spectrum} (DGS for short) if for any graph $H$, $H$ and $G$ are cospectral with cospectral complements implies that $H$ is isomorphic to $G$. In \cite{WX,WX1}, Wang and Xu gave some methods for determining whether a family of graphs are DGS. In this paper, we shall review some of the old results and present some new ones along this line of research.