On the connectivity of pp-diamond-free vertex transitive graphs

Let GG be a graph of order n(G)n(G), minimum degree δ(G)δ(G) and connectivity κ(G)κ(G). We call the graph GGmaximally connected   when κ(G)=δ(G)κ(G)=δ(G). The graph GG is said to be superconnected if every minimum vertex cut isolates a vertex.For an integer p≥1p≥1, we define a pp-diamond   as the graph with p+2p+2 vertices, where two adjacent vertices have exactly pp common neighbors, and the graph contains no further edges. Usually, the 1-diamond is triangle and the 2-diamond is diamond  . We call a graph pp-diamond-free   if it contains no pp-diamond as a (not necessarily induced) subgraph. A graph is vertex transitive if its automorphism group acts transitively on its vertex set.In this paper, we give some sufficient conditions for vertex transitive graphs to be maximally connected. In addition, superconnected pp-diamond-free (1≤p≤31≤p≤3) vertex transitive graphs are characterized.