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arXiv · 2411.05304

A sharp upper bound on the spectral radius of $θ(1,3,3)$-free graphs with given size

Yuxiang Liu·Ligong Wang
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Abstract

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 $G\cong S_{\frac{m+4}{2},2}^{-}$.

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Yuxiang Liu, Ligong Wang. 2024-11-21. A sharp upper bound on the spectral radius of $θ(1,3,3)$-free graphs with given size. https://arxiv.org/abs/2411.05304

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