A lower bound on the number of vertices of a graph

Abstract

Abstract. In this note, we derive a lower bound for the number of vertices of a graph in terms of its diameter, d, connectivity zc and minimum degree p which is sharper than that of Watkins [1] by an amount 2(p -k).Let G be any finite, undirected graph with neither loops nor multiple edges. Let zz, p, k and d denote the number of vertices, minimum degree, connectivity and diameter of G respectively. Watkins [1] has proved that if k > 1, then zz > k(d -1) + 2. He has used Menger's theorem to obtain the above result. In this note we prove a theorem from which Watkins' result follows as a corollary. Our proof is simple and elementary. Moreover the lower bound we obtain is sharper than that of Watkins by the amount 2(p -k).

Maximum diameter of regular digraphs

“…We should mention that Kane and Mohanty [6] show that if r is the minimum degree of the graph and d 2 L. Babai raised this question in connection with his work [l] (cf. [2, Section 61).…”