Degree-Similar Graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914872244043776 |
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| author | Godsil, Chris Sun, Wanting |
| author_facet | Godsil, Chris Sun, Wanting |
| contents | The degree matrix of a graph is the diagonal matrix with diagonal entries equal to the degrees of the vertices of $X$. If $X_1$ and $X_2$ are graphs with respective adjacency matrices $A_1$ and $A_2$ and degree matrices $D_1$ and $D_2$, we say that $X_1$ and $X_2$ are degree similar if there is an invertible real matrix $M$ such that $M^{-1}A_1M=A_2$ and $M^{-1}D_1M=D_2$. If graphs $X_1$ and $X_2$ are degree similar, then their adjacency matrices, Laplacian matrices, unsigned Laplacian matrices and normalized Laplacian matrices are similar. We first show that the converse is not true. Then, we provide a number of constructions of degree-similar graphs. Finally, we show that the matrices $A_1-μD_1$ and $A_2-μD_2$ are similar over the field of rational functions $\mathbb{Q}(μ)$ if and only if the Smith normal forms of the matrices $tI-(A_1-μD_1)$ and $tI-(A_2-μD_2)$ are equal. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_11328 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Degree-Similar Graphs Godsil, Chris Sun, Wanting Combinatorics 05C50 The degree matrix of a graph is the diagonal matrix with diagonal entries equal to the degrees of the vertices of $X$. If $X_1$ and $X_2$ are graphs with respective adjacency matrices $A_1$ and $A_2$ and degree matrices $D_1$ and $D_2$, we say that $X_1$ and $X_2$ are degree similar if there is an invertible real matrix $M$ such that $M^{-1}A_1M=A_2$ and $M^{-1}D_1M=D_2$. If graphs $X_1$ and $X_2$ are degree similar, then their adjacency matrices, Laplacian matrices, unsigned Laplacian matrices and normalized Laplacian matrices are similar. We first show that the converse is not true. Then, we provide a number of constructions of degree-similar graphs. Finally, we show that the matrices $A_1-μD_1$ and $A_2-μD_2$ are similar over the field of rational functions $\mathbb{Q}(μ)$ if and only if the Smith normal forms of the matrices $tI-(A_1-μD_1)$ and $tI-(A_2-μD_2)$ are equal. |
| title | Degree-Similar Graphs |
| topic | Combinatorics 05C50 |
| url | https://arxiv.org/abs/2407.11328 |