General Form of the Automorphism Group of Bicyclic Graphs

Abstract

In 1869, Jordan proved that the set $\mathcal{T}$ of all finite group that can be represented as the automorphism group of a tree is containing the trivial group and it is closed under taken direct product of groups of lower order in $\mathcal{T}$ and wreath product of a member in $\mathcal{T}$ and the symmetric group on $n$ symbols. The aim of this paper is to continue this work and another works by Klav$\acute{\rm i}$k and Zeman in 2017 to present a class $\mathcal{S}$ of finite groups for which the automorphism group of each bicyclic graph is a member of $\mathcal{S}$ and this class is minimal with this property.