Diameter and Connectivity of Finite Simple Graphs

Abstract

Let G be a finite simple non-complete connected graph on [n] = {1, . . . , n} and κ(G) ≥ 1 its vertex connectivity. Let f (G) denote the number of free vertices of G and diam(G) the diameter of G. The final goal of this paper is to determine all sequences of integers (n, f, d, k) with n ≥ 8, f ≥ 0, d ≥ 2 and k ≥ 1 for which there exists a finite simple non-complete connected graph on [n] with f = f (G), d = diam(G) and k = κ(G).