Spectral radius of graphs of given size with forbidden a fan graph $F_6$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916530680233984 |
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| author | Gao, Jing Li, Xueliang |
| author_facet | Gao, Jing Li, Xueliang |
| contents | Let $F_k=K_1\vee P_{k-1}$ be the fan graph on $k$ vertices. A graph is said to be $F_k$-free if it does not contain $F_k$ as a subgraph. Yu et al. in [arXiv:2404.03423] conjectured that for $k\geq2$ and $m$ sufficiently large, if $G$ is an $F_{2k+1}$-free or $F_{2k+2}$-free graph, then $λ(G)\leq \frac{k-1+\sqrt{4m-k^2+1}}{2}$ and the equality holds if and only if $G\cong K_k\vee\left(\frac{m}{k}-\frac{k-1}{2}\right)K_1$. Recently, Li et al. in [arXiv:2409.15918] showed that the above conjecture holds for $k\geq 3$. The only left case is for $k=2$, which corresponds to $F_5$ or $F_6$. Since the case of $F_5$ was solved by Yu et al. in [arXiv:2404.03423] and Zhang and Wang in [On the spectral radius of graphs without a gem, Discrete Math. 347 (2024) 114171]. So, one needs only to deal with the case of $F_6$. In this paper, we solve the only left case by determining the maximum spectral radius of $F_6$-free graphs with size $m\geq 88$, and the corresponding extremal graph. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_13792 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral radius of graphs of given size with forbidden a fan graph $F_6$ Gao, Jing Li, Xueliang Combinatorics 05C35, 05C50 Let $F_k=K_1\vee P_{k-1}$ be the fan graph on $k$ vertices. A graph is said to be $F_k$-free if it does not contain $F_k$ as a subgraph. Yu et al. in [arXiv:2404.03423] conjectured that for $k\geq2$ and $m$ sufficiently large, if $G$ is an $F_{2k+1}$-free or $F_{2k+2}$-free graph, then $λ(G)\leq \frac{k-1+\sqrt{4m-k^2+1}}{2}$ and the equality holds if and only if $G\cong K_k\vee\left(\frac{m}{k}-\frac{k-1}{2}\right)K_1$. Recently, Li et al. in [arXiv:2409.15918] showed that the above conjecture holds for $k\geq 3$. The only left case is for $k=2$, which corresponds to $F_5$ or $F_6$. Since the case of $F_5$ was solved by Yu et al. in [arXiv:2404.03423] and Zhang and Wang in [On the spectral radius of graphs without a gem, Discrete Math. 347 (2024) 114171]. So, one needs only to deal with the case of $F_6$. In this paper, we solve the only left case by determining the maximum spectral radius of $F_6$-free graphs with size $m\geq 88$, and the corresponding extremal graph. |
| title | Spectral radius of graphs of given size with forbidden a fan graph $F_6$ |
| topic | Combinatorics 05C35, 05C50 |
| url | https://arxiv.org/abs/2412.13792 |