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Essential connectivity and spectral radius of graphs

Wenxiu Ding, Dan Li, Yu Wang, Jixiang Meng

Abstract

A graph is trivial if it contains one vertex and no edges. The essential connectivity $κ^{\prime}$ of $G$ is defined to be the minimum number of vertices of $G$ whose removal produces a disconnected graph with at least two non-trivial components. Let $\mathcal{A}_n^{κ',δ}$ be the set of graphs of order $n$ with minimum degree $δ$ and essential connectivity $κ'$. In this paper, we determine the graphs attaining the maximum spectral radii among all graphs in $\mathcal{A}_n^{κ',δ}$ and characterize the corresponding extremal graphs. In addition, we also determine the digraphs which achieve the maximum spectral radii among all strongly connected digraphs with given essential connectivity and give the exact values of the spectral radii of these digraphs.

Essential connectivity and spectral radius of graphs

Abstract

A graph is trivial if it contains one vertex and no edges. The essential connectivity of is defined to be the minimum number of vertices of whose removal produces a disconnected graph with at least two non-trivial components. Let be the set of graphs of order with minimum degree and essential connectivity . In this paper, we determine the graphs attaining the maximum spectral radii among all graphs in and characterize the corresponding extremal graphs. In addition, we also determine the digraphs which achieve the maximum spectral radii among all strongly connected digraphs with given essential connectivity and give the exact values of the spectral radii of these digraphs.

Paper Structure

This paper contains 3 sections, 10 theorems, 42 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm1']}
  3. Proof of Theorem \ref{['thm2']}

Key Result

Theorem 1

Let $G$ be a connected graph of order $n\geq\kappa'+4$ with minimum degree $\delta$ and essential connectivity $\kappa'$. Then with equality if and only if $G\cong G_n^{\kappa',\delta}$.

Figures (1)

  • Figure 1: $\vec{G}_n^{k,m}$

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Claim 1
  • Claim 2
  • Claim 3
  • Lemma 5
  • ...and 3 more