The $M$-matrix group inverse problem for distance-biregular graphs

TL;DR

This paper derives the group inverse of the Laplacian matrix for distance-biregular graphs using equilibrium measures, providing new insights into their structure and conditions for the inverse to be an M-matrix.

Contribution

It explicitly computes the group inverse of the Laplacian for distance-biregular graphs and links equilibrium measures to the graph's properties, offering a novel characterization.

Findings

Explicit formula for the group inverse using equilibrium measures

Characterization of when the inverse is an M-matrix

Connection between equilibrium arrays and graph structure

Abstract

In this paper we provide the group inverse of the combinatorial Laplacian matrix of distance-biregular graphs using the so-called equilibrium measures for sets obtained by deleting a vertex. We also show that the two equilibrium arrays characterizing a distance-biregular graph can be expressed in terms of the mentioned equilibrium measures. As a consequence of the minimum principle, we show a characterization of when the group inverse of the combinatorial Laplacian matrix of a distance-biregular graph is an $M$-matrix.