Abstract

An Euclidean graph associated with a molecule is defined by a weighted graph with adjacency matrix M = [dij], where for i ≠ j, dij is the Euclidean distance between the nuclei i and j. In this matrix dii can be taken as zero if all the nuclei are equivalent. Otherwise, one may introduce different weights for distinct nuclei. In this paper we present some MATLAB and GAP programs and use them to find the automorphism group of the Bis Benzene Chromium(0) with D6d point group symmetry and the big fullerene C80.

AMS Subject Classification: 92E10.

Keywords: Euclidean graph, symmetry, Bis Benzene Chromium(0), fullerene.

Introduction

Let G = (V, E) be a simple graph. G is called a weighted graph if each edge e is assigned a non-negative number w(e), called the weight of e. The Euclidean graph of a molecule is a complete weighted graph in which the edges are weighted by the Euclidean distances of vertices.

An automorphism of a weighted graph G is a permutation g of the vertex set of G with the property that, (i) for any vertices u and v, g(u) and g(v) are adjacent if and only if u is adjacent to v; (ii) for every edge e, w(g(e)) = w(e). The set of all automorphisms of a weighted graph G, with the operation of composition of permutations, is a permutation group on V(G), denoted Aut(G).