Unicyclic graphs with maximum Randić indices.

The Randić index R(G) of a graph G is the sum of the weights (dudv)1/2 of all edges uv in G, where du denotes the degree of vertex u. Du and Zhou [On Randić indices of trees, unicyclic graphs, and bicyclic graphs, Int. J. Quantum Chem. 111 (2011), 2760-2770] determined the n-vertex unicyclic graphs with the third maximum for n ≥ 5, the fourth maximum for n ≥ 7 and the fifth maximum for n ≥ 8. Recently, Li et al. [The Randić indices of trees, unicyclic graphs and bicyclic graphs, Ars Comb. 127 (2016), 409-419] obtained the n-vertex unicyclic graphs with the sixth maximum and the seventh maximum for n ≥ 9 and the eighth maximum for n ≥ 10. In this paper, we characterize the n-vertex unicyclic graphs with the ninth maximum, the tenth maximum, the eleventh maximum, the twelfth maximum and the thirteenth maximum of Randić values. [ABSTRACT FROM AUTHOR]