Graphs with few distinct \(D\)-eigenvalues determined by their \(D\)-spectra
type: Item
instance of: scholarly article
title: Graphs with few distinct \(D\)-eigenvalues determined by their \(D\)-spectra
zbMATH Open document ID: 1485.05107
published in: Linear Algebra and its Applications
publication date: 2021-09-02T00:00:00Z
review text: The distance matrix \(D\) of a graph \(G\) with vertex set \(\{v_1,v_2,\ldots ,v_n\}\) is the \(n\times n\) matrix that has its \((i,j)\) position equal to the distance between the vertices \(v_i\) and \(v_j\). In this article, the author studies graphs whose distance matrix has at most three eigenvalues different than \(-1\) and \(-2\). He proves that if \(G\) is such a graph, then \(G\) can be obtained from either \(P_4\) or a connected graph \(H\) with at most \(3\) vertices, by replacing each vertex with either a clique or an independent set. If this is the case, then \(G\) is said to be a mixed extension of \(P_4\) or of \(H\). Furthermore, they characterized which graphs are determined by the spectrum of their distance matrix, and which are co-spectral. The main tools used are Cauchy's interlace theorem and the fact that if \(G\) is a mixed extension of \(H\), then the eigenvalues of \(G\) can be obtained from an \(|H|\times |H|\) matrix.
reviewed by: Adrián Pastine
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 05C12
zbMATH DE Number: 7390148
zbMATH Keywords: \(D\)-spectrum
zbMATH Keywords: \(D\)-cospectral graph
zbMATH Keywords: mixed extension
author: Xiaoling Zhang
MaRDI profile type: Publication
full work available at URL: j.laa.2021.06.017
OpenAlex ID: W3182799943
cites work: Q3907599
cites work: Which graphs are determined by their spectrum?
cites work: Graphs with at most three distance eigenvalues different from \(-1\) and \(-2\)
cites work: Complete multipartite graphs are determined by their distance spectra
cites work: On the distance spectrum of graphs
cites work: The graphs with exactly two distance eigenvalues different from \(-1\) and \(-3\)
cites work: Graphs with small diameter determined by their $D$-spectra
DOI: 10.1016/J.LAA.2021.06.017
description: scientific article; zbMATH DE number 7390148