Asymmetric Graphs

We consider in this paper only non-directed graphs without multiple edges and without loops . The number of vertices of a graph G will be called its order, and will be denoted by N(G) . We shall call such a graph symmetric, if there exists a non-identical permutation of its vertices, which leaves the graph invariant. By other words a graph is called symmetric if the group of its automorphisms has degree greater than 1 . A graph which is not symmetric will be called asymmetric . The degree of symmetry of a symmetric graph is evidently measured by the degree of its group of automorphisms . The question which led us to the results contained in the present paper is the following : how can we measure the degree of asymmetry of an asymmetric graph? Evidently any asymmetric graph can be made symmetric by deleting certain of its edges and by adding certain new edges connecting its vertices . We shall call such a transformation of the graph its symmetrization. For each symmetrization of the graph let us take the sum of the number of deleted edges say r and the number of new edges say s ; it is reasonable to define the degree of asymmetry A [G] of a graph G, as the minimum of r+s where the minimum is taken over all possible symmetrizations of the graph G. (In what follows if in order to make a graph symmetric we delete r of its edges and add s new edges, we shall say that we changed r + s edges.) Clearly the asymmetry of a symmetric graph is according to this definition equal to 0, while the asymmetry of any asymmetric graph is a positive integer . The question arises : how large can be the degree of asymmetry of a graph of order n (i . e. a graph which has n vertices)? We shall denote by A (n) the maximum of A [G] for all graphs G of order n (n = 2, 3, . . .) . We put further A (1) = + -. It is evident that A (2) =A (3) =0 . Now let G denote the complementary graph of G, that is the graph which consists of the same vertices as G and of those and only those edges which do not belong to G then we have evidently