On the first extended zeroth-order connectivity index of trees

The first extended zeroth-order connectivity index of a graph G is defined as 0 1/2 1 ( ) ( ) , v v V G ? G D? ? ? ? where V (G) is the vertex set of G, and v D is the sum of degrees of neighbors of vertex v in G. We give a sharp lower bound for the first extended zeroth-order connectivity index of trees with given numbers of vertices and pendant vertices, and characterize the extremal trees. We also determine the n-vertex trees with the first three smallest first extended zeroth-order connectivity indices.