Spectral extrema of graphs of given even size forbidding H(4,3)
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914052081451008 |
|---|---|
| author | Zheng, Ruiling Zhang, Gang |
| author_facet | Zheng, Ruiling Zhang, Gang |
| contents | A graph is sad to be $H$-free if it does not contain $H$ as a subgraph. Let $H(k,3)$ be the graph formed by taking a cycle of length $k$ and a triangle on a common vertex. Li, Lu and Peng [Discrete Math. 346 (2023) 113680] proved that if $G$ is an $H(3,3)$-free graph of size $m \geq 8$, then the spectral radius $ρ(G) \leq \frac{1+\sqrt{4 m-3}}{2}$ with equality if and only if $G \cong S_{\frac{m+3}{2}, 2}$, where $S_{\frac{m+3}{2}, 2}=K_2 \vee \frac{m-1}{2}K_1$. Note that the bound is attainable only when $m$ is odd. Recently, Pirzada and Rehman [Comput. Appl. Math. 44 (2025) 295] proved that if $G$ is an $\{H(3,3),H(4,3)\}$-free graph of even size $m \geq 10$, then $ρ(G) \leq ρ^{\prime}(m)$ with equality if and only if $G \cong S_{\frac{m+4}{2}, 2}^{-}$, where $ρ^{\prime}(m)$ is the largest root of $x^4-m x^2-(m-2) x+\frac{m}{2}-1=0$, and $S_{\frac{m+4}{2}, 2}^{-}$ is the graph obtained from $S_{\frac{m+4}{2}, 2}$ by deleting an edge incident to a vertex of degree two. In this paper, we improve the result of Pirzada and Rehman by showing that if $G$ is an $H(4,3)$-free graph of even size $m \geq 38$ without isolated vertices, then $ρ(G) \leq ρ^{\prime}(m)$ with equality if and only if $G \cong S_{\frac{m+4}{2}, 2}^{-}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_18594 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral extrema of graphs of given even size forbidding H(4,3) Zheng, Ruiling Zhang, Gang Combinatorics 05C50, 05C12, 15A18 A graph is sad to be $H$-free if it does not contain $H$ as a subgraph. Let $H(k,3)$ be the graph formed by taking a cycle of length $k$ and a triangle on a common vertex. Li, Lu and Peng [Discrete Math. 346 (2023) 113680] proved that if $G$ is an $H(3,3)$-free graph of size $m \geq 8$, then the spectral radius $ρ(G) \leq \frac{1+\sqrt{4 m-3}}{2}$ with equality if and only if $G \cong S_{\frac{m+3}{2}, 2}$, where $S_{\frac{m+3}{2}, 2}=K_2 \vee \frac{m-1}{2}K_1$. Note that the bound is attainable only when $m$ is odd. Recently, Pirzada and Rehman [Comput. Appl. Math. 44 (2025) 295] proved that if $G$ is an $\{H(3,3),H(4,3)\}$-free graph of even size $m \geq 10$, then $ρ(G) \leq ρ^{\prime}(m)$ with equality if and only if $G \cong S_{\frac{m+4}{2}, 2}^{-}$, where $ρ^{\prime}(m)$ is the largest root of $x^4-m x^2-(m-2) x+\frac{m}{2}-1=0$, and $S_{\frac{m+4}{2}, 2}^{-}$ is the graph obtained from $S_{\frac{m+4}{2}, 2}$ by deleting an edge incident to a vertex of degree two. In this paper, we improve the result of Pirzada and Rehman by showing that if $G$ is an $H(4,3)$-free graph of even size $m \geq 38$ without isolated vertices, then $ρ(G) \leq ρ^{\prime}(m)$ with equality if and only if $G \cong S_{\frac{m+4}{2}, 2}^{-}$. |
| title | Spectral extrema of graphs of given even size forbidding H(4,3) |
| topic | Combinatorics 05C50, 05C12, 15A18 |
| url | https://arxiv.org/abs/2509.18594 |