Neighborhood version of third Zagreb index of trees

Abstract

For a graph $G$, the third neighborhood degree index of $G$ is defined as: $$ND_3(G)=\sum_{ab\in E(G)}\delta_G(a)\delta_G(b)\Big(\delta_G(a)+\delta_G(b)\Big),$$ 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