On the maximum Graovac-Pisanski index of bicyclic graphs

Abstract

<abstract><p>For a simple graph $ G = (V(G), E(G)) $, the Graovac-Pisanski index of $ G $ is defined as</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ GP(G) = \frac{|V(G)|}{2|{\rm{Aut}}(G)|}\sum\limits_{u\in V(G)}\sum\limits_{\alpha\in {\rm{Aut}}(G)}d_G(u,\alpha(u)), $\end{document} </tex-math></disp-formula></p> <p>where $ {\rm{Aut}}(G) $ is the automorphism group of $ G $ and $ d_G(u, v) $ is the length of a shortest path between the two vertices $ u $ and $ v $ in $ G $. Obviously, $ GP(G) = 0 $ if $ G $ has no nontrivial automorphisms. Let $ B_{n}^{3, 3} $ be the graph consisting of two disjoint 3-cycles with a path of length $ n-5 $ joining them. In this article, we prove that among all those $ n $-vertex bicyclic graphs in which every edge lies on at most one cycle, $ B_{n}^{3, 3} $ has the maximum Graovac-Pisanski index.</p></abstract>