A combinatorial bound on the number of distinct eigenvalues of a graph
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866918250742284288 |
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| author | Allred, Sarah Erickson, Craig Grace, Kevin Hall, H. Tracy Jensen, Alathea |
| author_facet | Allred, Sarah Erickson, Craig Grace, Kevin Hall, H. Tracy Jensen, Alathea |
| contents | The smallest possible number of distinct eigenvalues of a graph $G$, denoted by $q(G)$, has a combinatorial bound in terms of unique shortest paths in the graph. In particular, $q(G)$ is bounded below by $k$, where $k$ is the number of vertices of a unique shortest path joining any pair of vertices in $G$. Thus, if $n$ is the number of vertices of $G$, then $n-q(G)$ is bounded above by the size of the complement (with respect to the vertex set of $G$) of the vertex set of the longest unique shortest path joining any pair of vertices of $G$. The purpose of this paper is to commence the study of the minor-monotone floor of $n-k$, which is the minimum of $n-k$ among all graphs of which $G$ is a minor. Accordingly, we prove some results about this minor-monotone floor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_11307 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A combinatorial bound on the number of distinct eigenvalues of a graph Allred, Sarah Erickson, Craig Grace, Kevin Hall, H. Tracy Jensen, Alathea Combinatorics 05C50, 15A18, 05C38, 05C83, 05C75, 05C85 The smallest possible number of distinct eigenvalues of a graph $G$, denoted by $q(G)$, has a combinatorial bound in terms of unique shortest paths in the graph. In particular, $q(G)$ is bounded below by $k$, where $k$ is the number of vertices of a unique shortest path joining any pair of vertices in $G$. Thus, if $n$ is the number of vertices of $G$, then $n-q(G)$ is bounded above by the size of the complement (with respect to the vertex set of $G$) of the vertex set of the longest unique shortest path joining any pair of vertices of $G$. The purpose of this paper is to commence the study of the minor-monotone floor of $n-k$, which is the minimum of $n-k$ among all graphs of which $G$ is a minor. Accordingly, we prove some results about this minor-monotone floor. |
| title | A combinatorial bound on the number of distinct eigenvalues of a graph |
| topic | Combinatorics 05C50, 15A18, 05C38, 05C83, 05C75, 05C85 |
| url | https://arxiv.org/abs/2209.11307 |