Subdivision-free graphs with the maximum spectral radius
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918085063081984 |
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| author | Sun, Wanting Wang, Guanghui Yang, Pingchuan |
| author_facet | Sun, Wanting Wang, Guanghui Yang, Pingchuan |
| contents | Given a graph family $\mathbb{H}$, let ${\rm SPEX}(n,\mathbb{H}_{\rm sub})$ denote the set of $n$-vertex $\mathbb{H}$-subdivision-free graphs with the maximum spectral radius. In this paper, we investigate the problem of graph subdivision from a spectral extremal perspective, with a focus on the structural characterization of graphs in ${\rm SPEX}(n,\mathbb{H}_{\rm sub})$. For any graph $H \in \mathbb{H}$, let $α(H)$ denote its independence number. Define $γ_\mathbb{H}:=\min_{H\in \mathbb{H}}\{|H| - α(H) - 1\}$. We prove that every graph in ${\rm SPEX}(n,\mathbb{H}_{\rm sub})$ contains a spanning subgraph isomorphic to $K_{γ_\mathbb{H}}\vee (n-γ_\mathbb{H})K_1$, which is obtained by joining a $γ_\mathbb{H}$-clique with an independent set of $n-γ_\mathbb{H}$ vertices. This extends a recent result by Zhai, Fang, and Lin concerning spectral extremal problems for $\mathbb{H}$-minor-free graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_04257 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Subdivision-free graphs with the maximum spectral radius Sun, Wanting Wang, Guanghui Yang, Pingchuan Combinatorics Given a graph family $\mathbb{H}$, let ${\rm SPEX}(n,\mathbb{H}_{\rm sub})$ denote the set of $n$-vertex $\mathbb{H}$-subdivision-free graphs with the maximum spectral radius. In this paper, we investigate the problem of graph subdivision from a spectral extremal perspective, with a focus on the structural characterization of graphs in ${\rm SPEX}(n,\mathbb{H}_{\rm sub})$. For any graph $H \in \mathbb{H}$, let $α(H)$ denote its independence number. Define $γ_\mathbb{H}:=\min_{H\in \mathbb{H}}\{|H| - α(H) - 1\}$. We prove that every graph in ${\rm SPEX}(n,\mathbb{H}_{\rm sub})$ contains a spanning subgraph isomorphic to $K_{γ_\mathbb{H}}\vee (n-γ_\mathbb{H})K_1$, which is obtained by joining a $γ_\mathbb{H}$-clique with an independent set of $n-γ_\mathbb{H}$ vertices. This extends a recent result by Zhai, Fang, and Lin concerning spectral extremal problems for $\mathbb{H}$-minor-free graphs. |
| title | Subdivision-free graphs with the maximum spectral radius |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.04257 |