The Randié Indices of Trees, Unicyclic Graphs and Bicyclic Graphs

Abstract

The Randić index $R(G)$ of a graph $G$ is the sum of the weights $(d_u{d}_v{)}^{-\frac{1}{2}}$ over all edges $uv$ of $G$, where $d_u$ denotes the degree of the vertex $u$. In this paper, we determine the first ten, eight, and six largest values for the Randić indices among all trees, unicyclic graphs, and bicyclic graphs of order $n\ge 11$, respectively. These extend the results of Du and Zhou [On Randić indices of trees, unicyclic graphs, and bicyclic graphs, International Journal of Quantum Chemistry, 111 (2011), 2760–2770].