Trees with a given order and matching number that have maximum general Randić index

The general Randić index R α ( G ) of a graph G is defined by R α ( G ) = ∑ u v ( d ( u ) d ( v ) ) α , where d ( u ) is the degree of a vertex u , and the summation extends over all edges u v of G . Some results on trees with a given order and matching number that have minimum general Randić index have been obtained. However, the corresponding maximum problem has not been studied, and usually the maximum problem is much harder than the minimum one. In this paper, we characterize the structure of the trees with a given order and matching number that have maximum general Randić index for α > 1 and give a sharp upper bound for 0 < α ≤ 1 .