Sharp lower bounds for the general Randić index of trees with a given size of matching

The general Randić index wα(G) of a graph G is the sum of the weights (d(u)d(v)) of all edges uv of G, where α is a real number and d(u) denotes the degree of the vertex u. Let T n,m be the set of all trees on n vertices with a maximum matching of cardinality m. Denote by T 0 n,m the tree on n vertices obtained from the star graph Sn−m+1 by attaching a pendant edge to each of some m − 1 non-central vertices of Sn−m+1. In this paper, we first prove that T 0 n,m has the minimum general Randić index among the trees in T n,m for − 12 ≤ α < 0. Also we obtain lower bounds for the general Randić index among trees in T n,m (2m ≤ n ≤ 3m+ 1) for α > 0, and the corresponding extremal graphs.