On the rank of the distance matrix of graphs

Let G be a connected graph with V (G ) = { v (1) , . . . ,v(n)}. The (i, j)-entry of the distance matrix D(G) of G is the distance between v(i) and v(j). 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 eta >= 1 there exists a trivially perfect graph with nullity eta. We also show that for threshold graphs, which are a subfamily of the family of trivially perfect graphs, the nullity is bounded by one. (C) 2022 Elsevier Inc. All rights reserved.