A Survey on Symmetry Group of Polyhedral Graphs

Modjtaba Ghorbani,Mina Rajabi-Parsa,Matthias Dehmer,Shaghayegh Rahmani

doi:10.3390/sym12030370

Abstract

Every three-connected simple planar graph is a polyhedral graph and a cubic polyhedral graph with pentagonal and hexagonal faces is called as a classical fullerene. The aim of this paper is to survey some results about the symmetry group of cubic polyhedral graphs. We show that the order of symmetry group of such graphs divides 240.

Highlights

In the present work, all graphs are connected without loops and parallel edges, which we call simple graphs
All three connected cubic planar graphs with hexagons and pentagons are called as fullerenes and we donote them by PH-fullerenes, see [2,3]
A three-connected cubic planar graph whose faces are triangles and hexagons is denoted by a TH-fullerene and a SH-fullerene is a three connected cubic planar graph with quadrangles and hexagons

Summary

Introduction

All graphs are connected without loops and parallel edges, which we call simple graphs. A fullerene is a cubic three-connected graph whose faces are pentagons and hexagons. We denote a cubic polyhedral graph with t triangles, s quadrangles, p pentagonal and h hexagonal faces and no other faces by a (t, s, p, h)−polyhedral or briefly a (t, s, p)-polyhedral graph. The cube graph Q3 is the smallest SPH-polyhedral graph which has no hexagonal face, see Figure 1a. The symmetry group and the point-group symmetry of a graph are not isomorphic but about the regular polehedral graphs they are the same By usung this fact, the authors of [30] computed the symmetry of all fullerenes with up to 70 vertices. The first author computed the automorphism group of some infinite families of fullerene graphs by using GAP programs [34]. This paper bears some novel results significant for algorithmization of the fullerene graph machine analysis

Main Results

Leapfrog Operation

The Symmetry Group of Non-Classical Fullerenes