On the connectivity and superconnected graphs with small diameter

In this paper, first we prove that any graph GG is 2-connected if diam(G)≤g−1diam(G)≤g−1 for even girth gg, and for odd girth gg and maximum degree Δ≤2δ−1Δ≤2δ−1 where δδ is the minimum degree. Moreover, we prove that any graph GG of diameter diam(G)≤g−2diam(G)≤g−2 satisfies that (i) GG is 5-connected for even girth gg and Δ≤2δ−5Δ≤2δ−5, and (ii) GG is super-κκ for odd girth gg and Δ≤3δ/2−1Δ≤3δ/2−1.