The maximum spectral radius of planner graphs without the joint of K2 and a linear forest

Weilun Xu; An Chang

paper

The maximum spectral radius of planner graphs without the joint of K2 and a linear forest

arXiv:2403.06383

Abstract

Given a graph $F$ , let $S P E X_{P} (n, F)$ be the set of graphs with the maximum spectral radius among all $F$ -free $n$ -vertex planner graph. In 2017, Tait and Tobin proved that for sufficiently $n$ , $K_{2} + P_{n - 2}$ is the unique graph with the maximum spectral radius over all $n$ -vertex planner graphs. In this paper, focusing on $S P E X_{P} (n, K_{2} + H)$ in which $H$ is a linear forest, we prove that $S P E X_{P} (n, K_{2} + H) = {2 K_{1} + C_{n - 2}}$ when $H \in {p K_{2}, P_{3}, I_{q}}$ $(p \geq 1, q \geq 3)$ , where $K_{n}$ , $P_{n}$ , $I_{n}$ are complete graph, path and empty graph of order $n$ , respectively. When $H$ contains a $P_{4}$ , we prove that $2 K_{1} + C_{n - 2} \in / S P E X_{P} (n, K_{2} + H)$ and also provide a structural characterization of graphs in $S P E X_{P} (n, K_{2} + H)$ .

11pages, 2 figures