A sharp upper bound on the spectral radius of $θ(1,3,3)$-free graphs with given size
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917843275087872 |
|---|---|
| author | Liu, Yuxiang Wang, Ligong |
| author_facet | Liu, Yuxiang Wang, Ligong |
| contents | A graph $G$ is $F$-free if $G$ does not contain $F$ as a subgraph. Let $ρ(G)$ be the spectral radius of a graph $G$. Let $θ(1,p,q)$ denote the theta graph, which is obtained by connecting two distinct vertices with three internally disjoint paths with lengths $1, p, q$, where $p\leq q$. Let $S_{n,k}$ denote the graph obtained by joining every vertex of $K_{k}$ to $n-k$ isolated vertices and $S_{n,k}^{-}$ denote the graph obtained from $S_{n,k}$ by deleting an edge incident to a vertex of degree $k$, respectively. In this paper, we show that if $ρ(G)\geqρ(S_{\frac{m+4}{2},2}^{-})$ for a graph $G$ with even size $m\geq 92$, then $G$ contains a $θ(1,3,3)$ unless $G\cong S_{\frac{m+4}{2},2}^{-}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_05304 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A sharp upper bound on the spectral radius of $θ(1,3,3)$-free graphs with given size Liu, Yuxiang Wang, Ligong Combinatorics A graph $G$ is $F$-free if $G$ does not contain $F$ as a subgraph. Let $ρ(G)$ be the spectral radius of a graph $G$. Let $θ(1,p,q)$ denote the theta graph, which is obtained by connecting two distinct vertices with three internally disjoint paths with lengths $1, p, q$, where $p\leq q$. Let $S_{n,k}$ denote the graph obtained by joining every vertex of $K_{k}$ to $n-k$ isolated vertices and $S_{n,k}^{-}$ denote the graph obtained from $S_{n,k}$ by deleting an edge incident to a vertex of degree $k$, respectively. In this paper, we show that if $ρ(G)\geqρ(S_{\frac{m+4}{2},2}^{-})$ for a graph $G$ with even size $m\geq 92$, then $G$ contains a $θ(1,3,3)$ unless $G\cong S_{\frac{m+4}{2},2}^{-}$. |
| title | A sharp upper bound on the spectral radius of $θ(1,3,3)$-free graphs with given size |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.05304 |