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.

确定确保图的哈密顿性的某些条件非常重要和有价值，因为确定图是否是哈密顿量是一个 NP 完全问题。 对于具有顶点集V ( G ) 和边集E ( G )的图G，第一个萨格勒布索引 ( \(M_{1}\) ) 和第二个萨格勒布索引 ( \(M_{2}\) ) 定义为\(M_{1}(G)=\sum \limits _{v_{ i}v_{j}\in E(G)}(d_{G}(v_{i})+d_{G}(v_{j}))\)和\(M_{2}(G)=\总和 \limits _{v_{i}v_{j}\in E(G)}d_{G}(v_{i})d_{G}(v_{j})\)，其中\(d_{G} (v_{i})\)表示V(G)\) 中顶点 \(v_{i}\) 的度数。G的萨格勒布指数之差 ( \(\Delta M\) )定义为\(\Delta M(G)=M_{2}(G)-M_{1}(G)\)。在本文中，我们试图寻找结构图论和化学图论之间的关系。对于\(\Delta M(G)\)，我们获得了一些充分条件，使图成为k哈密尔顿图、可追踪图、k边哈密尔顿图、k连通图、汉密尔顿连通图或k路径可覆盖图。