The spectral extrema of graphs of odd size forbidding $H(4,3)$ beyond the book graph
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| Format: | Preprint |
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2026
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| _version_ | 1866915772530425856 |
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| author | Wani, Abdul Basit Pirzada, S. Rehman, Amir |
| author_facet | Wani, Abdul Basit Pirzada, S. Rehman, Amir |
| contents | A graph is said to be $H$-free if it does not contain a subgraph isomorphic to $H$. The fish graph, denoted by $H(4, 3)$, is a $6-$vertex graph obtained from a cycle of length $4$ and a triangle by sharing a common vertex. Earlier it is shown that $λ(G)\leq \frac{1+\sqrt{4m-3}}{2}$ holds for all $H(4,3)-$free graphs of odd size $m\geq 44,$ and the equality holds if and only if $G\cong S_{\frac{m+3}{2},2},$ where $S_{\frac{m+3}{2},2}$ is the $m-$edge book graph $K_2 \vee \frac{m-1}{2}K_1,$ where $K_2 \vee \frac{m-1}{2}K_1,$ denotes the join of $K_2$ and $\frac{m-1}{2}K_1.$ Let $\mathcal{G}(m,H(4,3))$ denote the family of $H(4,3)$-free graphs with $m$ edges and no isolated vertices. We write $ \mathcal{G}(m,H(4,3)) \setminus \left\{ K_2 \vee \tfrac{m-1}{2}K_1 \right\} $ for the corresponding subfamily obtained by excluding the book graph. In this paper, we establish a sharp upper bound on the spectral radius of graphs over $\mathcal{G}(m,H(4,3))\setminus \{K_2 \vee \frac{m-1}{2}K_1\}$ for odd $m\geq 58$ and characterize the unique extremal graph attaining this bound. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_03861 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The spectral extrema of graphs of odd size forbidding $H(4,3)$ beyond the book graph Wani, Abdul Basit Pirzada, S. Rehman, Amir Combinatorics 05C50, 05C12, 15A18 A graph is said to be $H$-free if it does not contain a subgraph isomorphic to $H$. The fish graph, denoted by $H(4, 3)$, is a $6-$vertex graph obtained from a cycle of length $4$ and a triangle by sharing a common vertex. Earlier it is shown that $λ(G)\leq \frac{1+\sqrt{4m-3}}{2}$ holds for all $H(4,3)-$free graphs of odd size $m\geq 44,$ and the equality holds if and only if $G\cong S_{\frac{m+3}{2},2},$ where $S_{\frac{m+3}{2},2}$ is the $m-$edge book graph $K_2 \vee \frac{m-1}{2}K_1,$ where $K_2 \vee \frac{m-1}{2}K_1,$ denotes the join of $K_2$ and $\frac{m-1}{2}K_1.$ Let $\mathcal{G}(m,H(4,3))$ denote the family of $H(4,3)$-free graphs with $m$ edges and no isolated vertices. We write $ \mathcal{G}(m,H(4,3)) \setminus \left\{ K_2 \vee \tfrac{m-1}{2}K_1 \right\} $ for the corresponding subfamily obtained by excluding the book graph. In this paper, we establish a sharp upper bound on the spectral radius of graphs over $\mathcal{G}(m,H(4,3))\setminus \{K_2 \vee \frac{m-1}{2}K_1\}$ for odd $m\geq 58$ and characterize the unique extremal graph attaining this bound. |
| title | The spectral extrema of graphs of odd size forbidding $H(4,3)$ beyond the book graph |
| topic | Combinatorics 05C50, 05C12, 15A18 |
| url | https://arxiv.org/abs/2602.03861 |