Algebraic Connectivity and Disjoint Vertex Subsets of Graphs

Yan Sun,Faxu Li

doi:10.1155/2020/5763218

Abstract

It is well known that the algebraic connectivity of a graph is the second small eigenvalue of its Laplacian matrix. In this paper, we mainly research the relationships between the algebraic connectivity and the disjoint vertex subsets of graphs, which are presented through some upper bounds on algebraic connectivity.

Highlights

Let h(u, A) be the distance between the node u and A, which is the shortest distance of the node u ∈ N to a node of the set A
A result on the algebraic connectivity and two partitions of graphs is presented by Alon [22] and Milman [20] below
Where g 0, and since the coordinates of the center of gravity of the three regions are the average of the triangle region, the vectors g(u, A) + g(u, B) + g(u, C) 0

Summary

Introduction

Graph theory has provided chemists with a variety of useful tools, such as in the topological structure [1,2,3]. e Laplacian matrix of a graph G is denoted by L, and L D − A, where D is a diagonal matrix whose diagonal entries are its degrees and A is the adjacency matrix of G. e Laplacian eigenvalues of a graph G are the eigenvalues of L, denoted by 0 μN ≤ μN− 1 ≤ . . . ≤ μ1, which are all real and nonnegative. e second smallest Laplacian eigenvalue μN− 1 of a graph is well known as the algebraic connectivity, which was first studied by Fiedler [4]. e algebraic connectivity [5] of a graph is important for the connectivity of a graph [6], which can be used to measure the robustness of a graph. E algebraic connectivity [5] of a graph is important for the connectivity of a graph [6], which can be used to measure the robustness of a graph. E second smallest Laplacian eigenvalue μN− 1 of a graph is well known as the algebraic connectivity, which was first studied by Fiedler [4]. It has been emerged as an important parameter in many system problems [7,8,9,10,11,12,13,14,15,16,17,18]. The relationships are researched between the algebraic connectivity and disjoint vertex subsets of graphs, which are presented through some upper bounds

Preliminaries

Main Result