On general Zagreb indices and some Hamiltonian properties of graphs

Let G = (V, E) be a graph.For a vertex w V , the degree of w is denoted by d(w).The -first Zagreb index and the second Zagreb index of G are defined, respectively, as uV d (u) and uvE d (u)d (v).In the case of the -first Zagreb index, is a real number different from 0 and 1; for the -second Zagreb index, is a nonzero real number.The -first and -second Zagreb indices generalize the classical first and second Zagreb indices, respectively.In this paper, upper bounds are presented for the -first Zagreb index with > 1 and the -second Zagreb index with > 0.Moreover, sufficient conditions are established, based on either the -first Zagreb index with > 1 or the -second Zagreb index with > 0, for the existence of certain Hamiltonian properties in a graph.