Study on geometric–arithmetic, arithmetic–geometric and Randić indices of graphs

Topological indices are mathematical descriptors used in the field of chemistry to characterize the topological structure of chemical compounds. The Randić index ($R$), the geometric–arithmetic index ($GA$), and the arithmetic–geometric index ($AG$) represent three widely recognized topological indices. In most scenarios, the properties of $AG$ and $GA$ exhibit opposing tendencies. Furthermore, it is observed that, $AG\left(G\right)>R\left(G\right)$ and $GA\left(G\right)>R\left(G\right)$ for any given graph $G$. Our focus is thus directed towards investigating the gaps between $AG$ and $R$, as well as $GA$ and $R$. We find that the invariants $AG-R$ and $GA-R$ correlate well with some molecular properties. Numerous upper and lower bounds for the quantities $AG-R$ and $GA-R$ are computed for general graphs, bipartite graphs, chemical graphs, trees, and chemical trees, in terms of graph order, with an emphasis on characterizing extremal graphs.