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Maximum spread of $K_r$-minor free graphs

Wenyan Wang, Lele Liu, Yi Wang

Abstract

The spread of a graph is the difference between the largest and smallest eigenvalue of its adjacency matrix. In this paper, we investigate spread problems for graphs with excluded clique-minors. We show that for sufficiently large $n$, the $n$-vertex $K_r$-minor free graph with maximum spread is the join of a clique and an independent set, with $r-2$ and $n-r+2$ vertices, respectively.

Maximum spread of $K_r$-minor free graphs

Abstract

The spread of a graph is the difference between the largest and smallest eigenvalue of its adjacency matrix. In this paper, we investigate spread problems for graphs with excluded clique-minors. We show that for sufficiently large , the -vertex -minor free graph with maximum spread is the join of a clique and an independent set, with and vertices, respectively.

Paper Structure

This paper contains 7 sections, 17 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Useful results
  5. Preliminary results on $K_r$-minor free graphs
  6. Structure of $K_r$-minor free graphs with maximum spread
  7. The extremal graph of $K_{r}$-minor free graphs with maximum spread

Key Result

Theorem 1.1

Let $G$ be a $K_r$-minor free graph of order $n$. For $r\geq3$ and $n$ sufficiently large, we have with equality if and only if $G\cong K_{r-2}\vee(n-r+2)K_1$.

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 2.1: M2
  • Theorem 2.2: TA3
  • Theorem 2.3: T
  • Lemma 2.1: Cioaba-Feng-Tait-Zhang2020
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • ...and 20 more