Unicyclic graphs with the maximal value of Graovac-Pisanski index

Abstract

Let G be a graph and let Γ be its group of automorphisms. Graovac-Pisanski index of G is GP(G) = |V(G)| / (2|Γ|) ∑_{u ∈ V(G)} ∑_{α ∈ Γ} d(u, α(u)), where d(u, v) is the distance from u to v in G. One can observe that GP(G) = 0 if G has no nontrivial automorphisms, but it is not known which graphs attain the maximum value of Graovac-Pisanski index. In this paper we show that among unicyclic graphs on n vertices the n-cycle attains the maximum value of Graovac-Pisanski index.