Walk Matrix-Based Upper Bounds on Generalized Cospectral Mates
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909681524408320 |
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| author | Raza, Muhammad Shabbir, Mudassir Abbas, Waseem |
| author_facet | Raza, Muhammad Shabbir, Mudassir Abbas, Waseem |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06927 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Walk Matrix-Based Upper Bounds on Generalized Cospectral Mates Raza, Muhammad Shabbir, Mudassir Abbas, Waseem Combinatorics Commutative Algebra 05C50 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. |
| title | Walk Matrix-Based Upper Bounds on Generalized Cospectral Mates |
| topic | Combinatorics Commutative Algebra 05C50 |
| url | https://arxiv.org/abs/2507.06927 |