Diameter and Connectivity of Finite Simple Graphs

Abstract

Let G be a finite simple non-complete connected graph on $\{1, \ldots , n\}$ and $\varkappa (G) \ge 1$ its vertex connectivity. Let f(G) denote the number of free vertices of G and ${\text {diam}}(G)$ the diameter of G. Being motivated by the computation of the depth of the binomial edge ideal of G, the possible sequences (n, q, f, d) of integers for which there is a finite simple non-complete connected graph G on $\{1, \ldots , n\}$ with $q = \varkappa (G), f = f(G), d = {\text {diam}}(G)$ satisfying $f + d = n + 2 - q$ will be determined. Furthermore, finite simple non-complete connected graphs G on $\{1, \ldots , n\}$ satisfying $f(G) + {\text {diam}}(G) = n + 2 - \varkappa (G)$ will be classified.