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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2501.14218 |
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| _version_ | 1866929759115542528 |
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| author | Zhang, Wenqian |
| author_facet | Zhang, Wenqian |
| contents | For a graph $G$, its spectral radius $ρ(G)$ is the largest eigenvalue of its adjacency matrix. Let $\mathcal{F}$ be a finite family of graphs with $\min_{F\in \mathcal{F}}χ(F)=r+1\geq3$, where $χ(F)$ is the chromatic number of $F$. Set $t=\max_{F\in\mathcal{F}}|F|$. Let $T(rt,r)$ be the Turán graph of order $rt$ with $r$ parts. Assume that some $F_{0}\subseteq\mathcal{F}$ is a subgraph of the graph obtained from $T(rt,r)$ by embedding a path or a matching in one part. Let ${\rm EX}(n,\mathcal{F})$ be the set of graphs with the maximum number of edges among all the graphs of order $n$ containing not any $F\in\mathcal{F}$. Simonovits \cite{S1,S2} gave general results on the graphs in ${\rm EX}(n,\mathcal{F})$. Let ${\rm SPEX}(n,\mathcal{F})$ be the set of graphs with the maximum spectral radius among all the graphs of order $n$ containing not any $F\in\mathcal{F}$. Motivated by the work of Simonovits, we characterize the specified structure of the graphs in ${\rm SPEX}(n,\mathcal{F})$ in this paper. Moreover, some applications are also included. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_14218 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral skeletons and applications Zhang, Wenqian Combinatorics For a graph $G$, its spectral radius $ρ(G)$ is the largest eigenvalue of its adjacency matrix. Let $\mathcal{F}$ be a finite family of graphs with $\min_{F\in \mathcal{F}}χ(F)=r+1\geq3$, where $χ(F)$ is the chromatic number of $F$. Set $t=\max_{F\in\mathcal{F}}|F|$. Let $T(rt,r)$ be the Turán graph of order $rt$ with $r$ parts. Assume that some $F_{0}\subseteq\mathcal{F}$ is a subgraph of the graph obtained from $T(rt,r)$ by embedding a path or a matching in one part. Let ${\rm EX}(n,\mathcal{F})$ be the set of graphs with the maximum number of edges among all the graphs of order $n$ containing not any $F\in\mathcal{F}$. Simonovits \cite{S1,S2} gave general results on the graphs in ${\rm EX}(n,\mathcal{F})$. Let ${\rm SPEX}(n,\mathcal{F})$ be the set of graphs with the maximum spectral radius among all the graphs of order $n$ containing not any $F\in\mathcal{F}$. Motivated by the work of Simonovits, we characterize the specified structure of the graphs in ${\rm SPEX}(n,\mathcal{F})$ in this paper. Moreover, some applications are also included. |
| title | Spectral skeletons and applications |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2501.14218 |