On the Graovac-Pisanski index of a graph

Abstract

Let $G$ be a graph.Its Graovac-Pisanski index is\newline
$$
\GP(G)=\frac{|V(G)|}{2|\Aut(G)|}\sum_{u\in V(G)}
\sum_{\alpha\in\Aut(G)}\dist(u,\alpha(u)),
$$
where $\Aut(G)$ is the group
of automorphisms of $G$, and its Wiener index,
$W(G)$, is the sum of all
distances in $G$.
In the class of trees (unicyclic graphs) on $n$
vertices we find those with
the maximum value of Graovac-Pisanski index.
We show that the inequality $\GP(G)\le W(G)$ is
not true in general, but it
is true for trees.