Automorphism of a graph

An automorphism of a graph G is an isomorphism of G onto itself. It is easy to... 

The automorphism of a graph is the isomorphic mapping of that graph on itself. For example ... 

The set of all automorphisms of a graph forms a group 32-34). If the graph is drawn so that it possesses some symmetry, every symmetry operation is in fact an automorphism. Therefore, graphs can be treated as if they possess the symmetry of the corresponding molecule. 

On the other hand, every natural number is the order of the automorphism group of a graph with connectivity number 1. 

This method can be used more generally to find what can be called the "unlabelled chromatic polynomial" of a graph — giving the number of ways of coloring in x colors the vertices of a graph when two colorings are equivalent if one is converted to the other by an automorphism of the graph. 

We remind that an automorphism of a simple graph is a permutation of the vertices preserving adjacencies between vertices. For plane graphs, we require also that faces are sent to faces but for 3-connected graphs this condition is redundant. Recall that Aut(G) denotes the group of automorphisms of G. 

Some - shape descriptors and -> centric indices contain information about symmetry, while -> WHIM symmetry, - Bertz complexity index, - indices of neighbourhood symmetry, and - symmetry number obtained by the automorphism group of a graph are explicitly related to the symmetry. 

An isomorphic mapping of a graph G onto itself is called an autamarpki m. Thus, an automorphism is a bijective mapping of V G) onto itself, which maintains the pair relationship Gy... 

An isomorphism of a graph with itself is called an automorphism. An automorphism can be represented by a permutation (mapping) which transforms a graph, laBclijig into another labeling and preserves the adjacency of the vertices. A permutation P represented in a two-row notation has the following form ... 

An orbit is the set of all atoms which are transformed from one into another by the actions of all automorphisms of a molecular graph. TTie set of all different orbits of a molecular graph G forms a partition of G. If P represents an automorphism then atom i is symmetric (topologically equivalent) with its image p/, and atonis i and p, belong to the same orbit. This symmetry relationship in the molecular graph may or may not be true for the three-dimensional molecular structure. 

In a series of papers Uchino used the matrix multiplication method for obtaining the canonical code and automorphisms of a molecular graph. He considered adjacency, distance, and open walks matrices in a series of efficient algorithms which offer the automorphism partition of graphs. 

The automorphism group of a graph (molecular structure) is composed of all permutations of vertices (atoms) which do not make or break any of the original connections between vertices. 

Sp = 1. Thus N(C) is the sum of the coefficients in the expression C. For a cycle index A we have, by definition, N(A) = 1, From what has been stated above, it follows that N(A B) will be the num- ber of superpositions of the two graphs. Moreover the "product" A B can be extended, by associativity, to any number of cycle indexes, and will again be the cycle index sum for the superpositions of many graphs. Hence if A- denotes, for brevity, the cycle index of the automorphism group of Tj -- our previous Z(C.) --then the number of superposed graphs is given by... 

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