Document Type

Article

Publication Date

3-1-2022

Published In

La Matematica

Abstract

Distance matrices of graphs were introduced by Graham and Pollack in 1971 to study a problem in communications. Since then, there has been extensive research on the distance matrices of graphs—a 2014 survey by Aouchiche and Hansen on spectra of distance matrices of graphs lists more than 150 references. In the last 10 years, variants such as the distance Laplacian, the distance signless Laplacian, and the normalized distance Laplacian matrix of a graph have been studied. After a brief description of the early history of the distance matrix and its motivating problem, this survey focuses on comparing and contrasting techniques and results for the four types of distance matrices. Digraphs are treated separately after the discussion of graphs, including discussion of similarities and differences between graphs and digraphs. New results are presented that complement existing results, including results for some the matrices on unimodality of characteristic polynomials for graphs, preservation of parameters by cospectrality for graphs, and bounds on spectral radii for digraphs.

Keywords

Distance matrix, Distance signless Laplacian, Distance Laplacian, Normalized distance Laplacian

Comments

This version of the article has been accepted for publication and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online.

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