Lectures on Matrices and Graphs
(MG 2019)

Title of the talk : Moore-Penrose inverse of some combinatorial matrices

Abstract :
In the first part we consider the incidence matrix of a distance regular graph. The class of distance regular graphs is an important class of regular graphs while studying several topics such as combinatorial designs, symmetric nets and Hadamard matrices. We first review the important properties of distance regular graphs. We then obtain a graph-theoretic description of the Moore–Penrose inverse of the incidence matrix of a distance regular graph. We illustrate our results using several classical examples of distance regular graphs. Families of distance transitive graphs, which are a subclass of distance regular graphs, are also considered. In the later part we consider the resistance matrix of a directed graph and prove analogs of several known results. The technique involves the description of the Moore-Penrose inverse of the Laplacian.