On the Randić Index and Extremal Cacti
The Randić index of a graph G is the sum of ((d(u))(d(v))) α over all edges uv of G, where d(v) denotes the degree of v in G, α≠0. Earlier in [Discrete Appl. Math. 156, No. 10, 1725–1735 (2008; Zbl 1152.05320)] A. Lin, R. Juo and Y. Zha provided a sharp lower bound for the Randić index of cacti with given number of pendant edges, in the case of α=-1 2. In this short note we seek to provide some results regarding the extremal cacti with respect to general Randić indices (i.e., 0≠α∈[-1,1]) for cacti with given number of vertices, pendant edges and cycles. We conjecture that the extremal cacti in this category must be in a special group, a formula for the Randić index of these special cacti is provided. More generally, our approach lead to a single inequality for any value of α, the verification of which will result in a simple proof of our conjecture for the specific value of α. As an application, characterizations of the extremal cacti for the weight (special case of the Randić index when α=1) with various restrictions can be immediately achieved.
Wang, Hua, Daniel Gray.
"On the Randić Index and Extremal Cacti."
Congressus Numerantium, 194: 87-96.