Extremal Results on ℓ-Connected Graphs or Pancyclic Graphs Based on Wiener-Type Indices

A graph of order n is called pancyclic if it contains a cycle of length y for every 3≤y≤n. The connectivity of an incomplete graph G, denoted by κ(G), is min{|W||WisavertexcutofG}. A graph G is said to be ℓ-connected if the connectivity κ(G)≥ℓ. The Wiener-type indices of a connected graph G are Wg(G)=∑{s,t}⊆V(G)g(dG(s,t)), where g(x) is a function and dG(s,t) is the distance in G between s and t. In this note, we first determine the minimum and maximum values of Wg(G) for ℓ-connected graphs. Then, we use the Wiener-type indices of graph G, the Wiener-type indices of complement graph G¯ with minimum degree δ(G)≥2 or δ(G)≥3 to give some sufficient conditions for connected graphs to be pancyclic. Our results generalize some existing results of several papers.