General Form of the Automorphism Group of Bicyclic Graphs

TL;DR

This paper characterizes the automorphism groups of bicyclic graphs, extending classical results on trees, and identifies a minimal class of groups that can serve as automorphism groups for such graphs.

Contribution

It introduces a minimal class of finite groups that precisely correspond to automorphism groups of bicyclic graphs, building on foundational work on trees.

Findings

Identifies a minimal class of groups for automorphism groups of bicyclic graphs.

Extends Jordan's classical results on automorphism groups of trees.

Provides structural insights into automorphism groups 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.