Spectra of variants of distance matrices of graphs and digraphs: a survey

TL;DR

This survey reviews the development and current state of spectral graph theory related to various distance matrices and their variants, highlighting recent results and differences between graphs and digraphs.

Contribution

It provides a comprehensive comparison of techniques and results for four types of distance matrices, including new findings on spectral properties and bounds.

Findings

New results on unimodality of characteristic polynomials for graphs

Results on cospectrality preserving parameters in graphs

Bounds on spectral radii for digraphs

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 ten 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.