On the spectral extremal problem of planar graphs
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arXiv
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2024
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| _version_ | 1866911784099643392 |
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| author | Wang, Xiaolong Huang, Xueyi Lin, Huiqiu |
| author_facet | Wang, Xiaolong Huang, Xueyi Lin, Huiqiu |
| contents | The spectral extremal problem of planar graphs has aroused a lot of interest over the past three decades. In 1991, Boots and Royle [Geogr. Anal. 23(3) (1991) 276--282] (and Cao and Vince [Linear Algebra Appl. 187 (1993) 251--257] independently) conjectured that $K_2 + P_{n-2}$ is the unique graph attaining the maximum spectral radius among all planar graphs on $n$ vertices, where $K_2 + P_{n-2}$ is the graph obtained from $K_2\cup P_{n-2}$ by adding all possible edges between $K_2$ and $P_{n-2}$. In 2017, Tait and Tobin [J. Combin. Theory Ser. B 126 (2017) 137--161] confirmed this conjecture for all sufficiently large $n$. In this paper, we consider the spectral extremal problem for planar graphs without specified subgraphs. For a fixed graph $F$, let $\mathrm{SPEX}_{\mathcal{P}}(n,F)$ denote the set of graphs attaining the maximum spectral radius among all $F$-free planar graphs on $n$ vertices. We describe a rough sturcture for the connected extremal graphs in $\mathrm{SPEX}_{\mathcal{P}}(n,F)$ when $F$ is a planar graph not contained in $K_{2,n-2}$. As applications, we determine the extremal graphs in $\mathrm{SPEX}_{\mathcal{P}}(n,W_k)$, $\mathrm{SPEX}_{\mathcal{P}}(n,F_k)$ and $\mathrm{SPEX}_{\mathcal{P}}(n,(k+1)K_2)$ for all sufficiently large $n$, where $W_k$, $F_k$ and $(k+1)K_2$ are the wheel graph of order $k$, the friendship graph of order $2k+1$ and the disjoint union of $k+1$ copies of $K_2$, respectively. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_16419 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the spectral extremal problem of planar graphs Wang, Xiaolong Huang, Xueyi Lin, Huiqiu Combinatorics 05C50 The spectral extremal problem of planar graphs has aroused a lot of interest over the past three decades. In 1991, Boots and Royle [Geogr. Anal. 23(3) (1991) 276--282] (and Cao and Vince [Linear Algebra Appl. 187 (1993) 251--257] independently) conjectured that $K_2 + P_{n-2}$ is the unique graph attaining the maximum spectral radius among all planar graphs on $n$ vertices, where $K_2 + P_{n-2}$ is the graph obtained from $K_2\cup P_{n-2}$ by adding all possible edges between $K_2$ and $P_{n-2}$. In 2017, Tait and Tobin [J. Combin. Theory Ser. B 126 (2017) 137--161] confirmed this conjecture for all sufficiently large $n$. In this paper, we consider the spectral extremal problem for planar graphs without specified subgraphs. For a fixed graph $F$, let $\mathrm{SPEX}_{\mathcal{P}}(n,F)$ denote the set of graphs attaining the maximum spectral radius among all $F$-free planar graphs on $n$ vertices. We describe a rough sturcture for the connected extremal graphs in $\mathrm{SPEX}_{\mathcal{P}}(n,F)$ when $F$ is a planar graph not contained in $K_{2,n-2}$. As applications, we determine the extremal graphs in $\mathrm{SPEX}_{\mathcal{P}}(n,W_k)$, $\mathrm{SPEX}_{\mathcal{P}}(n,F_k)$ and $\mathrm{SPEX}_{\mathcal{P}}(n,(k+1)K_2)$ for all sufficiently large $n$, where $W_k$, $F_k$ and $(k+1)K_2$ are the wheel graph of order $k$, the friendship graph of order $2k+1$ and the disjoint union of $k+1$ copies of $K_2$, respectively. |
| title | On the spectral extremal problem of planar graphs |
| topic | Combinatorics 05C50 |
| url | https://arxiv.org/abs/2402.16419 |