Smith Normal Form of Distance Matrix of Block Graphs

Abstract

A connected graph, whose blocks are all cliques (of possibly varying sizes), is called a block graph. Let $D(G)$ be its distance matrix. In this note, we prove that the Smith normal form of $D(G)$ is independent of the interconnection way of blocks and give an explicit expression for the Smith normal form in the case that all cliques have the same size, which generalize the results on determinants.