A sufficient condition for generalized spectral characterization of graphs with loops

TL;DR

This paper establishes a sufficient condition involving the walk matrix for uniquely characterizing graphs with loops using their generalized spectrum, extending prior work on simple graphs.

Contribution

It introduces a new sufficient condition based on the walk matrix's square-free determinant for graphs with loops, generalizing previous spectral characterization results.

Findings

Graphs with loops are characterized by their generalized spectrum if the walk matrix has a square-free determinant.

The paper proves a general result about symmetric integral matrices.

Extends spectral characterization conditions from simple graphs to graphs with loops.

Abstract

Sufficient conditions for a simple graph to be characterized up to isomorphism given its spectrum and the spectrum of its complement graph are known due to Wang and Xu. This note establishes a related sufficient condition in the presence of loops: if the walk matrix has square-free determinant, then the graph is characterized by its generalized spectrum. The proof includes a general result about symmetric integral matrices.