On the connectivity and superconnected graphs with small diameter

Abstract

In this paper, first we prove that any graph G is 2-connected if diam (G)≤ g-1 for even girth g, and for odd girth g and maximum degree $\Delta$ ≤ 2$\delta$-1 where $\delta$ is the minimum degree. Moreover, we prove that any graph G of diameter diam(G) ≤ g-2 satisfies that (i)G is 5-connected for even girth g and $\Delta$ ≤ 2$\delta$-5, and (ii)G is super-k for odd girth g and $\Delta$ ≤ 3$\delta$/2-1.