The First Zagreb Index Conditions for Some Hamiltonian Properties of Graphs

Let $G = (V, E)$ be a graph. The first Zagreb index of a graph G is defined as ∑u ∈ V d²(u), where $d(u)$ is the degree of vertex u in G. Using the P\'{o}lya-Szeg{o} inequality, we in this paper present the first Zagreb index conditions for some Hamiltonian properties of a graph and an upper bound for the first Zagreb index of a graph.