The maximum spectral radius of $θ_{1,3,3}$-free graphs with given size
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| Format: | Preprint |
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2024
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| _version_ | 1866914969340084224 |
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| author | Gao, Jing Li, Xueliang |
| author_facet | Gao, Jing Li, Xueliang |
| contents | A graph $G$ is said to be $F$-free if it does not contain $F$ as a subgraph. A theta graph, say $θ_{l_1,l_2,l_3}$, is the graph obtained by connecting two distinct vertices with three internally disjoint paths of length $l_1, l_2, l_3$, where $l_1\leq l_2\leq l_3$ and $l_2\geq2$. Recently, Li, Zhao and Zou [arXiv:2409.15918v1] characterized the $θ_{1,p,q}$-free graph of size $m$ having the largest spectral radius, where $q\geq p\geq3$ and $p+q\geq2k+1\geq7$, and proposed a problem on characterizing the graphs with the maximum spectral radius among $θ_{1,3,3}$-free graphs. In this paper, we consider this problem and determine the maximum spectral radius of $θ_{1,3,3}$-free graphs with size $m$ and characterize the extremal graph. Up to now, all the graphs in $\mathcal{G}(m,θ_{1,p,q})$ which have the largest spectral radius have been determined, where $q\geq p\geq 2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_07721 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The maximum spectral radius of $θ_{1,3,3}$-free graphs with given size Gao, Jing Li, Xueliang Combinatorics 05C35, 05C50 A graph $G$ is said to be $F$-free if it does not contain $F$ as a subgraph. A theta graph, say $θ_{l_1,l_2,l_3}$, is the graph obtained by connecting two distinct vertices with three internally disjoint paths of length $l_1, l_2, l_3$, where $l_1\leq l_2\leq l_3$ and $l_2\geq2$. Recently, Li, Zhao and Zou [arXiv:2409.15918v1] characterized the $θ_{1,p,q}$-free graph of size $m$ having the largest spectral radius, where $q\geq p\geq3$ and $p+q\geq2k+1\geq7$, and proposed a problem on characterizing the graphs with the maximum spectral radius among $θ_{1,3,3}$-free graphs. In this paper, we consider this problem and determine the maximum spectral radius of $θ_{1,3,3}$-free graphs with size $m$ and characterize the extremal graph. Up to now, all the graphs in $\mathcal{G}(m,θ_{1,p,q})$ which have the largest spectral radius have been determined, where $q\geq p\geq 2$. |
| title | The maximum spectral radius of $θ_{1,3,3}$-free graphs with given size |
| topic | Combinatorics 05C35, 05C50 |
| url | https://arxiv.org/abs/2410.07721 |