Automorphous

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

Remarks on the Group of Automorphism of a Free Topological Tree... 

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

Let M, V be two vertices of a tree. We say they are similar if there is an automorphism of the tree which maps u onto v. This relation of similarity is an equivalence relation and partitions the p vertices of the tree into equivalence classes. Let p be the number of equivalence classes. Similarly we say that two edges of the tree are similar if there is an automorphism which maps one onto the other. Let q be the number of equivalence classes of edges under this relation. A symmetric edge in a tree is an edge, uv say, such that there is an automorphism of the tree which interchanges u and v. Let s be the number of symmetric edges in a tree it is easy to see that s can only be 0 or 1. We then have the following theorem. 

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... 

The distinct superpositions are shown in Figure 6 together with their automorphism groups, from which we can verify the assertions made above about the sum of their cycle indexes. For... 

For our present purpose we shall need to retain much more information about these graphs. Specifically, we want to find the sum of the cycle indexes of their automorphism groups. This is still basically a Polya-type problem, for which we replace T(x) by the sum of the cycle indexes of rooted trees. If T denotes the set of rooted trees, then this cycle index sum can be written Z(T ). Note that we can always recover F(x) from Z( T) for since the sum of the coefficients in the cycle index is 1, we have only to replace each occurrence of 5j by x Each cycle index for a tree on n vertices then reduces to x". This result is general and applies to any cycle index sum. 

In the general problem of this type the figure generating function would be that for the allowable radicals, and the group will be the group of automorphisms of the frame, restricted to those atoms which do not enjoy their full valency within the frame. 

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. 

SheJ68 Sheehan, J. The number of graphs with a given automorphism group. Canad. J. Math. 20 (1968) 1068-1076. 

StoP71 Stockmeyer, P. K. Enumeration of graphs with prescribed automorphism group. Ph.D. Thesis, U. of Michigan 1971. 

WhiD75 White, D. E. Classifying patterns by automorphism group an operator theoretic approach. Discrete Math. 13 (1975) 277-295. 

WhiD75c White, D. E. counting patterns with a given automorphism group. Proc. Amer. Math. Soc. 47 (1975) 41-44. 

Proof Let B be an abelian variety and h B — B an automorphism of B. Then every connected component of Bh is either an isolated point or a translation of an abelian subvariety of positive dimension of B. In particular e(Bh) is the number of isolated points in Bh. For a cycle a of length n we have... 

Finite groups of automorphisms of surfaces and the Mathieu group. Invent. Math. 94... 

Classification of inequivalent subalgebras of the algebras p(l,3), p(1.3), c(l,3) within actions of different automorphism groups [including the groups P(l, 3), P(l, 3) and 0(1,3)] is already available [30]. Since we will concentrate on conformally invariant systems, it is natural to restrict our disscussion to the classification of subalgebras of c(l, 3) that are inequivalent within the action of the conformal group 0(1, 3). 

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