Sufficient conditions for hamiltonian properties of graphs based on the difference of Zagreb indices

A graph invariant, in the sense of graph automorphism, is a mapping from the set of graphs to the reals. Numerous topological indices have been proposed to characterize the topological properties of graphs, and they are widely recognized as graph invariants. Some sufficient conditions in terms of certain topological indices have been suggested to describe hamiltonian properties of graphs, such as Hamiltonicity, traceability, Hamiltonian-connectedness, k-leaf-connectedness, as well as \(\beta \)-deficiency. For a graph G, the first and second Zagreb indices are defined as \(M_1(G)= \sum \nolimits _{u\in V(G)} {d_u^2}\) and \(M_2(G)= \sum \nolimits _{uv\in E(G)} {d_u}{d_v}\), where \(d_u\) denotes the degree of vertex u in G. The difference of Zagreb indices of G is defined as \(\Delta M(G) = {M_2}(G) - {M_1}(G)\). In this paper, we suggest some sufficient conditions in terms of \(\Delta M(G)\) for graphs to be Hamiltonian, Hamiltonian-connected and \(\beta \)-deficient, respectively.