Some Inequalities between General Randić-Type Graph Invariants

Abstract

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 $R_{\alpha }\left(G\right)={{\displaystyle \sum }}_{{\upsilon }_i\sim {\upsilon }_j}{\left(d_i{d}_j\right)}^{\alpha }$ and $Q_{\alpha }\left(G\right)={{\displaystyle \sum }}_{v_i\in V}{d}_i^{\alpha }$, respectively, where $\alpha$ is an arbitrary real number, $d_i$ denotes the degree of a vertex ${\upsilon }_i$, and ${\upsilon }_i\sim {\upsilon }_j$ represents the adjacency of vertices ${\upsilon }_i$ and ${\upsilon }_j$ in $G$. Establishing relationships between two topological indices holds significant importance for researchers. Some implicit inequality relationships between $R_{\alpha }$ and $Q_{\alpha }$ have been derived so far. In this paper, we establish explicit inequality relationships between $R_{\alpha }$ and $Q_{\alpha }$. 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 $\alpha$.