Graphs that are cospectral for the distance Laplacian

2018/12/13 by Brimkov, Boris, Duna, Ken, Hogben, Leslie +4
#05C12 #05C500 #15A18 #15B57 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1812.05734

Abstract

The distance matrix D(G) of a graph G is the matrix containing the pairwise distances between vertices, and the distance Laplacian matrix is D^L(G)=T(G)-D(G), where T(G) is the diagonal matrix of row sums of D(G). We establish several general methods for producing D^L-cospectral graphs that can be used to construct infinite families. We provide examples showing that various properties are not preserved by D^L-cospectrality, including examples of D^L-cospectral strongly regular and circulant graphs. We establish that the absolute values of coefficients of the distance Laplacian characteristic polynomial are decreasing, i.e., |δ^L₁|≥ … ≥ |δ^L_n| where δ^L_k is the coefficient of x^k.

Related

• The normalized distance Laplacian
• Cospectral constructions for several graph matrices using cousin vertices
• A construction of distance cospectral graphs
• A cospectral construction for the generalized distance matrix
• Spectra of variants of distance matrices of graphs and digraphs: a survey