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 (M1\documentclass[12pt]{minimal}
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\begin{document}$$M_{1}$$\end{document}) and second Zagreb index (M2\documentclass[12pt]{minimal}
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\begin{document}$$M_{2}$$\end{document}) are defined as M1(G)=∑vivj∈E(G)(dG(vi)+dG(vj))\documentclass[12pt]{minimal}
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\begin{document}$$M_{1}(G)=\sum \limits _{v_{i}v_{j}\in E(G)}(d_{G}(v_{i})+d_{G}(v_{j}))$$\end{document} and M2(G)=∑vivj∈E(G)dG(vi)dG(vj)\documentclass[12pt]{minimal}
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\begin{document}$$M_{2}(G)=\sum \limits _{v_{i}v_{j}\in E(G)}d_{G}(v_{i})d_{G}(v_{j})$$\end{document}, where dG(vi)\documentclass[12pt]{minimal}
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\begin{document}$$d_{G}(v_{i})$$\end{document} denotes the degree of vertex vi∈V(G)\documentclass[12pt]{minimal}
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\begin{document}$$v_{i}\in V(G)$$\end{document}. The difference of Zagreb indices (ΔM\documentclass[12pt]{minimal}
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\begin{document}$$\Delta M$$\end{document}) of G is defined as ΔM(G)=M2(G)-M1(G)\documentclass[12pt]{minimal}
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\begin{document}$$\Delta M(G)=M_{2}(G)-M_{1}(G)$$\end{document}.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 ΔM(G)\documentclass[12pt]{minimal}
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\begin{document}$$\Delta M(G)$$\end{document}, for graphs to be k-hamiltonian, traceable, k-edge-hamiltonian, k-connected, Hamilton-connected or k-path-coverable.
