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 (\(M_{1}\)) and second Zagreb index (\(M_{2}\)) are defined as \(M_{1}(G)=\sum \limits _{v_{i}v_{j}\in E(G)}(d_{G}(v_{i})+d_{G}(v_{j}))\) and \(M_{2}(G)=\sum \limits _{v_{i}v_{j}\in E(G)}d_{G}(v_{i})d_{G}(v_{j})\), where \(d_{G}(v_{i})\) denotes the degree of vertex \(v_{i}\in V(G)\). The difference of Zagreb indices (\(\Delta M\)) of G is defined as \(\Delta M(G)=M_{2}(G)-M_{1}(G)\).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 \(\Delta M(G)\), for graphs to be k-hamiltonian, traceable, k-edge-hamiltonian, k-connected, Hamilton-connected or k-path-coverable.