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.