@article{Lu2023, 
author = {Jian Lu and Zhongxiang Wang},
title = {On the maximum Graovac-Pisanski index of bicyclic graphs},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {10},
pages = {24914-24928},
keywords = {modified Wiener index, Graovac-Pisanski index, automorphism, orbit, bicyclic graphs},
url = {https://www.sciopen.com/article/10.3934/math.20231270},
doi = {10.3934/math.20231270},
abstract = {For a simple graph    G  =  (  V  (  G  )  ,  E  (  G  )  ), the Graovac-Pisanski index of    G is defined as     G  P  (  G  )  =                    |            V      (      G      )              |                    2              |                              A          u          t                    (      G      )              |                  ∑          u      ∈      V      (      G      )            ∑          α      ∈                        A          u          t                    (      G      )            d    G    (  u  ,  α  (  u  )  )  ,where              A      u      t        (  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,    G  P  (  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.}
}