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arXiv · 1912.06335

Comparison of Wiener index and Zagreb eccentricity indices

Kexiang Xu·Kinkar Chandra Das·Sandi Klavžar·Huimin Li
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Abstract

The first and the second Zagreb eccentricity index of a graph $G$ are defined as $E_1(G)=\sum_{v\in V(G)}\varepsilon_{G}(v)^{2}$ and $E_2(G)=\sum_{uv\in E(G)}\varepsilon_{G}(u)\varepsilon_{G}(v)$, respectively, where $\varepsilon_G(v)$ 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.

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Kexiang Xu, Kinkar Chandra Das, Sandi Klavžar, Huimin Li. 2019-12-13. Comparison of Wiener index and Zagreb eccentricity indices. https://arxiv.org/abs/1912.06335

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