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

2024/03/11 by Weilun Xu, Xu, Weilun, An Chang +1 · 1 citation
Computer Science · Mathematics · #Advanced Graph Theory Research #Graph theory and applications #Graph Labeling and Dimension Problems

paper · pdf · doi:10.48550/arxiv.2403.06383

Abstract

Given a graph F, let SPEX_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, K₂+P_n-2 is the unique graph with the maximum spectral radius over all n-vertex planner graphs. In this paper, focusing on SPEX_P(n,K₂+H) in which H is a linear forest, we prove that SPEX_P(n,K₂+H)=\2K₁+C_n-2\ when H∈ \pK₂,P₃,I_q\ (p≥1, q≥ 3), where K_n, P_n, I_n are complete graph, path and empty graph of order n, respectively. When H contains a P₄, we prove that 2K₁+C_n-2∉ SPEX_P(n,K₂+H) and also provide a structural characterization of graphs in SPEX_P(n,K₂+H).

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