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

TL;DR

This paper establishes a sharp upper bound on the spectral radius of graphs that do not contain a specific theta subgraph, identifying the extremal graph structure for large even-sized graphs.

Contribution

It provides a new spectral bound for $ heta(1,3,3)$-free graphs and characterizes the extremal graphs achieving this bound.

Findings

Spectral radius bound for $ heta(1,3,3)$-free graphs

Identification of extremal graph $S_{(m+4)/2,2}^{-}$

Condition under which a graph contains $ heta(1,3,3)$

Abstract

A graph $G$ is $F$-free if $G$ does not contain $F$ as a subgraph. Let $\rho(G)$ be the spectral radius of a graph $G$. Let $\theta(1,p,q)$ 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 $\rho(G)\geq\rho(S_{\frac{m+4}{2},2}^{-})$ for a graph $G$ with even size $m\geq 92$, then $G$ contains a $\theta(1,3,3)$ unless $G\cong S_{\frac{m+4}{2},2}^{-}$.