On sufficient conditions for Hamiltonicity of graphs, and beyond

Identifying certain conditions that ensure the Hamiltonicity of graphs is highly important and valuable due to the fact that determining whether a graph is Hamiltonian is an NP-complete problem.For a graph G with vertex set V(G) and edge set E(G), the first Zagreb index ( $$M_{1}$$ M 1 ) and second Zagreb index ( $$M_{2}$$ M 2 ) are defined as $$M_{1}(G)=\sum \limits _{v_{i}v_{j}\in E(G)}(d_{G}(v_{i})+d_{G}(v_{j}))$$ M 1 ( G ) = ∑ v i v j ∈ E ( G ) ( d G ( v i ) + d G ( v j ) ) and $$M_{2}(G)=\sum \limits _{v_{i}v_{j}\in E(G)}d_{G}(v_{i})d_{G}(v_{j})$$ M 2 ( G ) = ∑ v i v j ∈ E ( G ) d G ( v i ) d G ( v j ) , where $$d_{G}(v_{i})$$ d G ( v i ) denotes the degree of vertex $$v_{i}\in V(G)$$ v i ∈ V ( G ) . The difference of Zagreb indices ( $$\Delta M$$ Δ M ) of G is defined as $$\Delta M(G)=M_{2}(G)-M_{1}(G)$$ Δ M ( G ) = M 2 ( G ) - M 1 ( G ) .In this paper, we try to look for the relationship between structural graph theory and chemical graph theory. We obtain some sufficient conditions, with regards to $$\Delta M(G)$$ Δ M ( G ) , for graphs to be k-hamiltonian, traceable, k-edge-hamiltonian, k-connected, Hamilton-connected or k-path-coverable.