Minimum higher eigenvalues of Laplacians on graphs
type: Item
instance of: scholarly article
title: Minimum higher eigenvalues of Laplacians on graphs
zbMATH Open document ID: 0858.05071
DOI: 10.1215/S0012-7094-96-08301-5
author: Joel Friedman
published in: Duke Mathematical Journal
publication date: 1997-03-23T00:00:00Z
review text: If \(G\) is an undirected graph, let \(A\) be the adjacency matrix of \(G\), \(D\) be the diagonal matrix whose \((v,v)\) entry is the degree of \(v\), and \(\Delta=D-A\) be the Laplacian of \(G\). If \(G\) is a connected graph, it is known that the eigenvalues of \(\Delta\) satisfy the condition \(0=\mu_1<\mu_2\leq\cdots\leq \mu_n\). The author finds the smallest possible \(i\)th eigenvalue \(\mu_i\) of \(\Delta\) in the class of all connected graphs on \(n\) vertices. He gives examples of graphs which achieve this value, and determines when the graph achieving this value is unique.
Mathematics Subject Classification ID: 05C50
zbMATH DE Number: 912118
zbMATH Keywords: adjacency matrix
zbMATH Keywords: diagonal matrix
zbMATH Keywords: Laplacian
zbMATH Keywords: eigenvalues
zbMATH Keywords: connected graphs
reviewed by: Aleksandar Torgašev
MaRDI profile type: Publication
cites work: State splitting for variable-length graphs (Corresp.)
cites work: Difference Equations, Isoperimetric Inequality and Transience of Certain Random Walks
cites work: Some geometric aspects of graphs and their eigenfunctions
cites work: Non-negative matrices and Markov chains. 2nd ed
full work available at URL: s0012-7094-96-08301-5
OpenAlex ID: W1997700207
description: scientific article; zbMATH DE number 912118