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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.06383 |
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| _version_ | 1866916155744059392 |
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| author | Xu, Weilun Chang, An |
| author_facet | Xu, Weilun Chang, An |
| contents | Given a graph $F$, let $SPEX_P(n,F)$ be the set of graphs with the maximum spectral radius among all $F$-free $n$-vertex planner graph. In 2017, Tait and Tobin proved that for sufficiently $n$, $K_2+P_{n-2}$ is the unique graph with the maximum spectral radius over all $n$-vertex planner graphs. In this paper, focusing on $SPEX_P(n,K_2+H)$ in which $H$ is a linear forest, we prove that $SPEX_P(n,K_2+H)=\{2K_1+C_{n-2}\}$ when $H\in \{pK_2,P_3,I_q\}$ $(p\geq1, q\geq 3)$, where $K_n$, $P_n$, $I_n$ are complete graph, path and empty graph of order $n$, respectively. When $H$ contains a $P_4$, we prove that $2K_1+C_{n-2}\notin SPEX_P(n,K_2+H)$ and also provide a structural characterization of graphs in $SPEX_P(n,K_2+H)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06383 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The maximum spectral radius of planner graphs without the joint of K2 and a linear forest Xu, Weilun Chang, An Combinatorics Given a graph $F$, let $SPEX_P(n,F)$ be the set of graphs with the maximum spectral radius among all $F$-free $n$-vertex planner graph. In 2017, Tait and Tobin proved that for sufficiently $n$, $K_2+P_{n-2}$ is the unique graph with the maximum spectral radius over all $n$-vertex planner graphs. In this paper, focusing on $SPEX_P(n,K_2+H)$ in which $H$ is a linear forest, we prove that $SPEX_P(n,K_2+H)=\{2K_1+C_{n-2}\}$ when $H\in \{pK_2,P_3,I_q\}$ $(p\geq1, q\geq 3)$, where $K_n$, $P_n$, $I_n$ are complete graph, path and empty graph of order $n$, respectively. When $H$ contains a $P_4$, we prove that $2K_1+C_{n-2}\notin SPEX_P(n,K_2+H)$ and also provide a structural characterization of graphs in $SPEX_P(n,K_2+H)$. |
| title | The maximum spectral radius of planner graphs without the joint of K2 and a linear forest |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.06383 |