Neighborhood version of third Zagreb index of trees

For a graph G, the third neighborhood degree index of G is defined as: ND_۳(G)=\sum_{ab\in E(G)}\delta_G(a)\delta_G(b)\Big(\delta_G(a)+\delta_G(b)\Big), where \delta_G(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.

Keywords:

Third neighborhood degree index‎ , ‎trees‎ , ‎lower bound‎