Wiener-type invariants on graph properties

The Wiener-type invariants of a simple connected graph G = (V(G), E(G)) can be expressed in terms of the quantities W_f =^∑{u,v}⊆V(G) f (d_G(u, v)) for various choices of the function f (x), where d_G(u, v) is the distance between vertices u and v in G. In this paper, we mainly give some sufficient conditions for a connected graph to be k-connected, β-deficient, k-hamiltonian, k-edge-hamiltonian, k-path-coverable or satisfy α(G) ≤ k.