neighborhood version of third zagreb index of trees

For a graph g, the third neighborhood degree index of g is defined as: nd3(g) = ∑ ab∈e(g) δg(a)δg(b) (δg(a) + δg(b)), where δg(a) represents the sum of degrees of all neighboring vertices of vertex a. in this short paper, we establish a new lower bound on the third neighborhood degree index of trees and characterize the extremal trees achieving this bound.