Radius, diameter, and minimum degree

We give asymptotically sharp upper bounds for the maximum diameter and radius of (i) a connected graph, {ii) a connected trangle-free graph, (iii) a connected C,-free graph with n vertices and with minimum degree 6, where n tends to infinity. Some conjectures for J&-free graphs are also stated. Let G be a connected graph with vertex set V(G) and edge set E(G). For any XJJ E V(G) let dG(x,y) denote the distance between x and y, i.e., the minimum length of an x-y path in G. The diameter and the radius of G are defined as The following theorem answers a question of Gallai [6]. THEOREM 1. Let G be a connected graph M*ith n vertices and with minimum degree 6 2 2. Then (ii) diam G< &-1.