Degree-similar graphs and cospectral graphs
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| Format: | Preprint |
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2025
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| author | Fan, Yi-Zheng Xing, Ruo-Jie Zhang, Yi-Liu Wang, Wei |
| author_facet | Fan, Yi-Zheng Xing, Ruo-Jie Zhang, Yi-Liu Wang, Wei |
| contents | Let $G$ be a graph with adjacency matrix $A(G)$ and degree matrix $D(G)$, and let $L_μ(G):=A(G)-μD(G)$. Two graphs $G_1$ and $G_2$ are called \emph{degree-similar} if there exists an invertible matrix $M$ such that $M^{-1} A(G_1) M =A(G_2)$ and $M^{-1} D(G_1) M =D(G_2)$. In this paper, we address three problems concerning degree-similar graphs proposed by Godsil and Sun. First, we present a new characterization of degree-similar graphs using degree partition, from which we derive methods and examples for constructing cospectral graphs and degree-similar graphs. Second, we construct infinite pairs of non-degree-similar trees $G_1$ and $G_2$ such that $tI- L_μ(G_1)$ and $tI-L_μ(G_2)$ have the same Smith normal form over $\Q(μ)[t]$, which provides a negative answer to a problem posed by Godsil and Sun. Third, we establish several invariants of degree-similar graphs and obtain results on unicyclic graphs that are degree-similar determined. Lastly we prove that for a strongly regular graph $G$ and any two edges $e$ and $f$ of $G$, $G \backslash e$ and $G \backslash f$ have identical $μ$-polynomial, i.e., $\det(tI-L_μ(G \backslash e))=\det(tI-L_μ(G \backslash f))$, which enables the construction of pairs of non-isomorphic graphs with same $μ$-polynomial, where $G \backslash e$ denotes the graph obtained from $G$ by deleting the edge $e$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_01520 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Degree-similar graphs and cospectral graphs Fan, Yi-Zheng Xing, Ruo-Jie Zhang, Yi-Liu Wang, Wei Combinatorics 05C50 Let $G$ be a graph with adjacency matrix $A(G)$ and degree matrix $D(G)$, and let $L_μ(G):=A(G)-μD(G)$. Two graphs $G_1$ and $G_2$ are called \emph{degree-similar} if there exists an invertible matrix $M$ such that $M^{-1} A(G_1) M =A(G_2)$ and $M^{-1} D(G_1) M =D(G_2)$. In this paper, we address three problems concerning degree-similar graphs proposed by Godsil and Sun. First, we present a new characterization of degree-similar graphs using degree partition, from which we derive methods and examples for constructing cospectral graphs and degree-similar graphs. Second, we construct infinite pairs of non-degree-similar trees $G_1$ and $G_2$ such that $tI- L_μ(G_1)$ and $tI-L_μ(G_2)$ have the same Smith normal form over $\Q(μ)[t]$, which provides a negative answer to a problem posed by Godsil and Sun. Third, we establish several invariants of degree-similar graphs and obtain results on unicyclic graphs that are degree-similar determined. Lastly we prove that for a strongly regular graph $G$ and any two edges $e$ and $f$ of $G$, $G \backslash e$ and $G \backslash f$ have identical $μ$-polynomial, i.e., $\det(tI-L_μ(G \backslash e))=\det(tI-L_μ(G \backslash f))$, which enables the construction of pairs of non-isomorphic graphs with same $μ$-polynomial, where $G \backslash e$ denotes the graph obtained from $G$ by deleting the edge $e$. |
| title | Degree-similar graphs and cospectral graphs |
| topic | Combinatorics 05C50 |
| url | https://arxiv.org/abs/2509.01520 |