A sufficient condition for generalized spectral characterization of graphs with loops

2025/11/24 by Van Werde, Alexander
#05C50 #11C20 #15B36 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2511.19625

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.

Citations

• Graph isomorphism and multivariate graph spectrum
• Switching methods of level 2 for the construction of cospectral graphs
• Smith Normal Form and the Generalized Spectral Characterization of Graphs
• Square-free Discriminants of Matrices and the Generalized Spectral Characterizations of Graphs
• A simple arithmetic criterion for graphs being determined by their generalized spectra
• Generalized Spectral Characterization of Graphs: Revisited
• A sufficient condition for a family of graphs being determined by their generalized spectra

Related

• Equivalence between spectral properties of graphs with and without loops
• Generalized Spectral Characterization of Graphs: Revisited
• New families of graphs determined by their generalized spectrum
• A simple arithmetic criterion for graphs being determined by their generalized spectra
• Smith Normal Form and the Generalized Spectral Characterization of Graphs