The Randić-type graph invariants are extensively investigated vertex-degree-based topological indices and have gained much prominence in recent years. The general Randić and zeroth-order general Randić indices are Randić-type graph invariants and are defined for a graph with vertex set as and , respectively, where is an arbitrary real number, denotes the degree of a vertex , and represents the adjacency of vertices and in . Establishing relationships between two topological indices holds significant importance for researchers. Some implicit inequality relationships between and have been derived so far. In this paper, we establish explicit inequality relationships between and . Also, we determine linear inequality relationships between these graph invariants. Moreover, we obtain some new inequalities for various vertex-degree-based topological indices by the appropriate choice of .

中文翻译：

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一般 Randić 型图不变量之间的一些不等式

Randić 型图不变量被广泛研究的基于顶点度的拓扑指数，并在近年来获得了广泛的关注。一般 Randić 和零阶一般 Randić 指数是 Randić 型图不变量，并且针对顶点设置为的图定义和分别为，其中是任意实数，表示顶点 的度，并且表示顶点和中的邻接关系。建立两个拓扑指数之间的关系对于研究人员来说具有重要意义。到目前为止，已经导出了和之间的一些隐含的不平等关系。在本文中，我们在和之间建立了明确的不等式关系。此外，我们还确定了这些图不变量之间的线性不等式关系。此外，我们通过适当选择 ，获得了各种基于顶点度的拓扑索引的一些新的不等式。