2025/07/09 by Muhammad Ali Raza, Raza, Muhammad, Mudassir Shabbir +3
Computer Science · Mathematics · #05C50 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2507.06927
openalex publication_date 2025/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The problem of characterizing graphs determined by their spectrum (DS) or generalized spectrum (DGS) has been a longstanding topic of interest in spectral graph theory, originating from questions in chemistry and mathematical physics. While previous studies primarily focus on identifying whether a graph is DGS, we address a related yet distinct question: how many non-isomorphic generalized cospectral mates a graph can have? Building upon recent advances that connect this question to the properties of the walk matrix, we introduce a broad family of graphs and establish an explicit upper bound on the number of non-isomorphic generalized cospectral mates they can have. This bound is determined by the arithmetic structure of the determinant of the walk matrix, offering a refined criterion for quantifying the multiplicity of generalized cospectral graphs. This result sheds new light on the structure of generalized cospectral graphs and provides a refined arithmetic criterion for bounding their multiplicity.