The Laplacian eigenvalues of graphs: a survey

TL;DR

This survey reviews the spectral properties of the Laplacian matrix of graphs, highlighting bounds, classifications, and applications across various graph types and invariants, while also presenting new results and open questions.

Contribution

It provides a comprehensive classification of Laplacian eigenvalue bounds for different graph classes and introduces new unpublished results and research questions.

Findings

Bounds for Laplacian eigenvalues depend on graph invariants.

Classifications for special graphs like trees and bipartite graphs.

New unpublished results and open questions are presented.

Abstract

The Laplacian matrix of a simple graph is the difference of the diagonal matrix of vertex degree and the (0,1) adjacency matrix. In the past decades, the Laplacian spectrum has received much more and more attention, since it has been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. This paper is primarily a survey of various aspects of the eigenvalues of the Laplacian matrix of a graph for the past teens. In addition, some new unpublished results and questions are concluded. Emphasis is given on classifications of the upper and lower bounds for the Laplacian eigenvalues of graphs (including some special graphs, such as trees, bipartite graphs, triangular-free graphs, cubic graphs, etc.) as a function of other graph invariants, such as degree sequence, the average 2-degree, diameter, the maximal independence number, the maximal matching number, vertex connectivity, the domination number, the number of the spanning trees, etc.