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arXiv · 2507.06927

Walk Matrix-Based Upper Bounds on Generalized Cospectral Mates

Muhammad Raza·Mudassir Shabbir·Waseem Abbas
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Abstract

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.

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Muhammad Raza, Mudassir Shabbir, Waseem Abbas. 2025-07-09. Walk Matrix-Based Upper Bounds on Generalized Cospectral Mates. https://arxiv.org/abs/2507.06927

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