The Randic 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, a ≠ 0. Earlier in Discrete Applied Mathematics, Lin, Luo and Zha provided a sharp lower bound for the Randic 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 Randic 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 Randic index of these special cacti is provided. More generally, our approach lead to a single inequality for any value of a, the verification of which will result in a simple proof of our conjecture for the specific value of a. As an application, characterizations of the extremal cacti for the weight (special case of the Randic index when α = 1) with various restrictions can be immediately achieved.
展开▼
