A sharp spectral extremal result for general non-bipartite graphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911316217692160 |
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| author | Byrne, John |
| author_facet | Byrne, John |
| contents | For a graph family $\mathcal F$, let $\mathrm{ex}(n,\mathcal F)$ and $\mathrm{spex}(n,\mathcal F)$ denote the maximum number of edges and maximum spectral radius of an $n$-vertex $\mathcal F$-free graph, respectively, and let $\mathrm{EX}(n,\mathcal F)$ and $\mathrm{SPEX}(n,\mathcal F)$ denote the corresponding sets of extremal graphs. Wang, Kang, and Xue showed that if $r\ge 2$ and $\mathrm{ex}(n,F)=e(T_{n,r})+O(1)$ then $\mathrm{SPEX}(n,\mathcal F)\subseteq\mathrm{EX}(n,\mathcal F)$ for $n$ large enough. Fang, Tait, and Zhai extended this result by showing if $e(T_{n,r})\le\mathrm{ex}(n,\mathcal F)<e(T_{n,r})+\lfloor n/2r\rfloor$ then $\mathrm{SPEX}(n,\mathcal F)\subseteq\mathrm{EX}(n,\mathcal F)$ for $n$ large enough, and asked for the maximum constant $c(r)$ such that $\mathrm{ex}(n,\mathcal F)\le e(T_{n,r})+(c(r)-\varepsilon)n$ guarantees such containment. In this paper we determine $c(r)$ exactly for all $r\ge 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18637 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A sharp spectral extremal result for general non-bipartite graphs Byrne, John Combinatorics 05C35 (Primary) 05C50 (Secondary) For a graph family $\mathcal F$, let $\mathrm{ex}(n,\mathcal F)$ and $\mathrm{spex}(n,\mathcal F)$ denote the maximum number of edges and maximum spectral radius of an $n$-vertex $\mathcal F$-free graph, respectively, and let $\mathrm{EX}(n,\mathcal F)$ and $\mathrm{SPEX}(n,\mathcal F)$ denote the corresponding sets of extremal graphs. Wang, Kang, and Xue showed that if $r\ge 2$ and $\mathrm{ex}(n,F)=e(T_{n,r})+O(1)$ then $\mathrm{SPEX}(n,\mathcal F)\subseteq\mathrm{EX}(n,\mathcal F)$ for $n$ large enough. Fang, Tait, and Zhai extended this result by showing if $e(T_{n,r})\le\mathrm{ex}(n,\mathcal F)<e(T_{n,r})+\lfloor n/2r\rfloor$ then $\mathrm{SPEX}(n,\mathcal F)\subseteq\mathrm{EX}(n,\mathcal F)$ for $n$ large enough, and asked for the maximum constant $c(r)$ such that $\mathrm{ex}(n,\mathcal F)\le e(T_{n,r})+(c(r)-\varepsilon)n$ guarantees such containment. In this paper we determine $c(r)$ exactly for all $r\ge 3$. |
| title | A sharp spectral extremal result for general non-bipartite graphs |
| topic | Combinatorics 05C35 (Primary) 05C50 (Secondary) |
| url | https://arxiv.org/abs/2411.18637 |