On sufficient conditions for Hamiltonicity of graphs, and beyond

Abstract

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.