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| Main Authors: | , , , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.12917 |
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| _version_ | 1866910705604624384 |
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| author | Barrett, Wayne Fallat, Shaun Furst, Veronika Nasserasr, Shahla Rooney, Brendan Tait, Michael |
| author_facet | Barrett, Wayne Fallat, Shaun Furst, Veronika Nasserasr, Shahla Rooney, Brendan Tait, Michael |
| contents | The parameter $q(G)$ of an $n$-vertex graph $G$ is the minimum number of distinct eigenvalues over the family of symmetric matrices described by $G$. We show that all $G$ with $e(\overline{G}) = |E(\overline{G})| \leq \lfloor n/2 \rfloor -1$ have $q(G)=2$. We conjecture that any $G$ with $e(\overline{G}) \leq n-3$ satisfies $q(G) = 2$. We show that this conjecture is true if $\overline{G}$ is bipartite and in other sporadic cases. Furthermore, we characterize $G$ with $\overline{G}$ bipartite and $e(\overline{G}) = n-2$ for which $q(G) > 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12917 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Graphs with Bipartite Complement that Admit Two Distinct Eigenvalues Barrett, Wayne Fallat, Shaun Furst, Veronika Nasserasr, Shahla Rooney, Brendan Tait, Michael Combinatorics 05C50, 15A29, 15A18 The parameter $q(G)$ of an $n$-vertex graph $G$ is the minimum number of distinct eigenvalues over the family of symmetric matrices described by $G$. We show that all $G$ with $e(\overline{G}) = |E(\overline{G})| \leq \lfloor n/2 \rfloor -1$ have $q(G)=2$. We conjecture that any $G$ with $e(\overline{G}) \leq n-3$ satisfies $q(G) = 2$. We show that this conjecture is true if $\overline{G}$ is bipartite and in other sporadic cases. Furthermore, we characterize $G$ with $\overline{G}$ bipartite and $e(\overline{G}) = n-2$ for which $q(G) > 2$. |
| title | Graphs with Bipartite Complement that Admit Two Distinct Eigenvalues |
| topic | Combinatorics 05C50, 15A29, 15A18 |
| url | https://arxiv.org/abs/2411.12917 |