The eccentric-distance sum of some graphs

Abstract

Let $G = (V,E)$ be a simple connected graph. The
eccentric-distance sum of $G$ is defined as
$\xi^{ds}(G) =\ds\sum_{\{u,v\}\subseteq V(G)} [e(u)+e(v)] d(u,v)$, where $e(u)$ %\ds
is the eccentricity of the vertex $u$ in $G$ and $d(u,v)$ is the
distance between $u$ and $v$. In this paper, we establish formulae
to calculate the eccentric-distance sum for some graphs, namely
wheel, star, broom, lollipop, double star, friendship, multi-star
graph and the join of $P_{n-2}$ and $P_2$.