Generalized spectral characterization of signed bipartite graphs

2025/05/18 by Songlin Guo, Wei Wang, Guo, Songlin +3
Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2505.12446

openalex publication_date 2025/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Σ be an n-vertex controllable or almost controllable signed bipartite graph, and let Δ_Σ denote the discriminant of its characteristic polynomial χ(Σ; x). We prove that if (\rmnum1) the integer 2 ^{-\lfloor n/2 \rfloor} √(Δ_Σ) is squarefree, and (\rmnum2) the constant term (even n) or linear coefficient (odd n) of χ(Σ; x) is ± 1, then Σ is determined by its generalized spectrum. This result extends a recent theorem of Ji, Wang, and Zhang [Electron. J. Combin. 32 (2025), #P2.18], which established a similar criterion for signed trees with irreducible characteristic polynomials.

Citations

• Primary decomposition theorem and generalized spectral characterization of graphs
• A new criterion for oriented graphs to be determined by their generalized skew spectrum
• Generalized spectral characterization of signed trees
• Smith Normal Form and the Generalized Spectral Characterization of Graphs
• Generalized spectral characterizations of almost controllable 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
• Controllable Subsets in Graphs
• A sufficient condition for a family of graphs being determined by their generalized spectra

Related

• Generalized spectral characterization of signed trees
• Spectral Fundamentals and Characterizations of Signed Directed Graphs
• The signed graphs with symmetric spectra
• Primary decomposition theorem and generalized spectral characterization of graphs
• Open problems in the spectral theory of signed graphs