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.

Density results for Graovac-Pisanski’s distance number

“…The GP index of truncation graphs, Thorn graphs, and caterpillars were calculated by Iranmanesh and Shabani [7]. Additionally, Knor et al [8] considered the maximum GP index among all graphs of a fixed order. Note that these results on the GP index have direct implications for the GP distance number.…”

Analysis of the Graovac–Pisanski Index of Some Polyhedral Graphs Based on Their Symmetry Group

“…Recently, some papers are devoted to finding the extremal graphs with respect to GP number, see [15,16]. In [17] it is proved that if T is a tree, then W(T) ≤ W(T) and in [18] the authors showed if G is either a connected bipartite graph or a connected graph of even order, then W(G) is an integer.…”