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 \(n \geq 11\), 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].