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

Weilun Xu, An Chang

TL;DR

This paper proves that 2K_1+C_{n-2}\notin SPEX_P(n,K_2+H) and provides a structural characterization of graphs in $SPEX_P(n,K_2+H)$, and provides a structural characterization of graphs in $SPEX_P(n,K_2+H)$.

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 $n$, $K_2+P_{n-2}$ is the unique graph with the maximum spectral radius over all $n$-vertex planner graphs. In this paper, focusing on $SPEX_P(n,K_2+H)$ in which $H$ is a linear forest, we prove that $SPEX_P(n,K_2+H)=\{2K_1+C_{n-2}\}$ when $H\in \{pK_2,P_3,I_q\}$ $(p\geq1, q\geq 3)$, 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 $2K_1+C_{n-2}\notin SPEX_P(n,K_2+H)$ and also provide a structural characterization of graphs in $SPEX_P(n,K_2+H)$.

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

TL;DR

This paper proves that 2K_1+C_{n-2}\notin SPEX_P(n,K_2+H) and provides a structural characterization of graphs in , and provides a structural characterization of graphs in .

Abstract

Given a graph , let be the set of graphs with the maximum spectral radius among all -free -vertex planner graph. In 2017, Tait and Tobin proved that for sufficiently , is the unique graph with the maximum spectral radius over all -vertex planner graphs. In this paper, focusing on in which is a linear forest, we prove that when , where , , are complete graph, path and empty graph of order , respectively. When contains a , we prove that and also provide a structural characterization of graphs in .
Paper Structure (3 sections, 8 theorems, 24 equations, 2 figures)

This paper contains 3 sections, 8 theorems, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of main results

Key Result

Theorem 1.1

Let $F$ be a connected planner graph with chromatic number $4$. Suppose that $n$ is large enough and $G \in SPEX_P(n,F)$. Then one of the following holds: (i) $G\cong2K_1+C_{n-2}$. (ii) $K_2+I_{n-2}$ is a spanning subgraph of $G$.

Figures (2)

  • Figure 1: The graph $K_2+P_{n-2}$.
  • Figure 2: The graph $2K_1+C_{8}$.

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition
  • Lemma 2.1