SUFFICIENT CONDITIONS FOR CERTAIN STRUCTURAL PROPERTIES OF GRAPHS BASED ON WIENER-TYPE INDICES

Let G = (V, E) be a simple connected graph with vertex set V and edge set E. The Wiener-type invariants of G = (V, E) can be expressed in terms of the quantities W-f = Sigma({u, v}subset of V) f (d(G) (u, v)), for various choices of the function f, where d(G)(u,v) is the distance between the vertices u and v in G. In this paper, we establish sufficient conditions based on Wiener-type indices under which every path of length r is contained in a Hamiltonian cycle and under which a bipartite graph on n + m, m > n, vertices contains a cycle of size 2n.