On the rank of the distance matrix of graphs

Abstract

Let G be a connected graph with V (G) = {v1, . . . , vn}. The (i, j)entry of the distance matrix D(G) of G is the distance between vi and vj. In this article, using the well-known Ramsey's theorem, we prove that for each integer k ≥ 2, there is a finite amount of graphs whose distance matrices have rank k. We exhibit the list of graphs with distance matrices of rank 2 and 3. Besides, we study the rank of the distance matrices of graphs belonging to a family of graphs with their diameters at most two, the trivially perfect graphs. We show that for each η ≥ 1 there exists a trivially perfect graph with nullity η. We also show that for threshold graphs, which are a subfamily of the family of trivially perfect graphs, the nullity is bounded by one.