Some Inequalities between General Randić-Type Graph Invariants

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 G with vertex set V as RαG=∑υi∼υjdidjα and QαG=∑vi∈Vdiα, respectively, where α is an arbitrary real number, di denotes the degree of a vertex υi, and υi∼υj represents the adjacency of vertices υi and υj in G. Establishing relationships between two topological indices holds significant importance for researchers. Some implicit inequality relationships between Rα and Qα have been derived so far. In this paper, we establish explicit inequality relationships between Rα and Qα. 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 α.