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Connected graphs minimizing the spectral radius for given order and dissociation number

Zejun Huang, Chenxi Yang

Abstract

A dissociation set in a graph is a subset of vertices which induces a subgraph with maximum degree at most one. The dissociation number of a graph is the maximum cardinality of its dissociation sets. In this paper, we consider the $n$-vertex connected graphs with a given dissociation number that attain the minimum spectral radius. By using structure analysis and constructing difference equations, we characterize the extremal graphs with dissociation number $n-3$.

Connected graphs minimizing the spectral radius for given order and dissociation number

Abstract

A dissociation set in a graph is a subset of vertices which induces a subgraph with maximum degree at most one. The dissociation number of a graph is the maximum cardinality of its dissociation sets. In this paper, we consider the -vertex connected graphs with a given dissociation number that attain the minimum spectral radius. By using structure analysis and constructing difference equations, we characterize the extremal graphs with dissociation number .
Paper Structure (3 sections, 15 theorems, 39 equations, 6 figures)

This paper contains 3 sections, 15 theorems, 39 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Priliminaries
  3. Proof of Theorem \ref{['theo1']}

Key Result

Theorem 2

Let $n\ge 39$ be an integer and $m=\lfloor {n}/{6}\rfloor$. If $G\in\mathcal{G}_{n,n-3}$ attains the minimum spectral radius, then

Figures (6)

  • Figure 1: $G(a,b,c;p,q,r)$
  • Figure 2: The graphs $W_n$, $E_6,~E_7,~E_8$
  • Figure 3: The graphs $\tilde{W}_{n}$, $\tilde{E}_6,\tilde{E}_7,\tilde{E}_8$
  • Figure 4: $G(a,b,c;p,q,r)$ with labeled Perron vector
  • Figure 5: $H(a,b,c;p,q,r)$
  • ...and 1 more figures

Theorems & Definitions (19)

  • Theorem 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • Lemma 11
  • ...and 9 more