The First Zagreb Index and Some Hamiltonian Properties of Graphs

Let G=(V,E) be a graph. The first Zagreb index of a graph G is defined as & sum;u is an element of VdG2(u), where dG(u) is the degree of vertex u in G. A graph G is called Hamiltonian (resp. traceable) if G has a cycle (resp. path) containing all the vertices of G. Using two established inequalities, in this paper, we present sufficient conditions involving the first Zagreb index for Hamiltonian graphs and traceable graphs. We also present upper bounds for the first Zagreb index of a graph and characterize the graphs achieving the upper bounds.