Comparison of Wiener index and Zagreb eccentricity indices

Abstract

The first and the second Zagreb eccentricity index of a graph $G$ are defined as $E_1(G)={\sum }_{v\in V(G)}{\epsilon }_G(v{)}^2$ and $E_2(G)={\sum }_{uv\in E(G)}{\epsilon }_G(u){\epsilon }_G(v)$, respectively, where ${\epsilon }_G(v)$ is the eccentricity of a vertex $v$. In this paper the invariants $E_1$, $E_2$, and the Wiener index are compared on graphs with diameter $2$, on trees, on a newly introduced class of universally diametrical graphs, and on Cartesian product graphs. In particular, if the diameter of a tree $T$ is not too big, then $W(T)\ge E_2(T)$ holds, and if the diameter of $T$ is large, then $W(T)<E_1(T)$ holds.

References in corpus (1)

• Eccentric Connectivity Index of Chemical Trees