The maximum spectral radius of $\theta_{1,3,3}$-free graphs with given size

TL;DR

This paper determines the maximum spectral radius of $ heta_{1,3,3}$-free graphs with a given size and characterizes the extremal graph, extending previous work on related theta-free graphs.

Contribution

It specifically solves the problem of finding the maximum spectral radius for $ heta_{1,3,3}$-free graphs and characterizes the extremal graph structure.

Findings

Identified the extremal $ heta_{1,3,3}$-free graph with maximum spectral radius.

Extended previous results on $ heta_{1,p,q}$-free graphs to the case $p=q=3.

Provided a complete characterization of the extremal graphs for given size.

Abstract

A graph $G$ is said to be $F$-free if it does not contain $F$ as a subgraph. A theta graph, say $\theta_{l_1,l_2,l_3}$, is the graph obtained by connecting two distinct vertices with three internally disjoint paths of length $l_1, l_2, l_3$, where $l_1\leq l_2\leq l_3$ and $l_2\geq2$. Recently, Li, Zhao and Zou [arXiv:2409.15918v1] characterized the $\theta_{1,p,q}$-free graph of size $m$ having the largest spectral radius, where $q\geq p\geq3$ and $p+q\geq2k+1\geq7$, and proposed a problem on characterizing the graphs with the maximum spectral radius among $\theta_{1,3,3}$-free graphs. In this paper, we consider this problem and determine the maximum spectral radius of $\theta_{1,3,3}$-free graphs with size $m$ and characterize the extremal graph. Up to now, all the graphs in $\mathcal{G}(m,\theta_{1,p,q})$ which have the largest spectral radius have been determined, where $q\geq p\geq 2$.