On Groups and Graphs(1)

By a graph X we mean a finite set V(X), called the vertices of X, together with a set E(X), called the edges of X, consisting of unordered pairs of distinct elements of V(X). We shall indicate the unordered pairs by brackets. Two graphs X and Y are said to be isomorphic, denoted by X cz Y, if there is a one-to-one map a of V(X) onto V(Y) such that [aa,ab~\eE(Y) if and only if [a, b~]eE(X). An isomorphism of X onto itself is said to be an automorphism of X. For each given graph X there is a group of automorphisms, denoted by G(A'), where the multiplication is the multiplication of permutations. The complementary graph Xe of X is the graph whose V(XC) = V(X), and E(XC) consists of all possible edges which do not belong to E(X). It is easy to see that X and Xe have the same group of automorphisms. A graph consisting of isolated vertices only is called the null graph, and its complementary graph is called the complete graph. Both the null graph and the complete graph of n vertices have S„, the symmetric group of n letters, as their group of automorphisms. A regular graph of degree k is a graph such that the number of edges incident with each vertex is k. The null graphs and the complete graphs are regular. The graph X is necessarily regular if G(X) is transitive. In [7], König proposed the following question: "When can a given abstract group be set up as the group of automorphisms of a graph?" The question can be interpreted in two ways, (a) Given a finite group G, can one construct a graph whose group of automorphisms is abstractly isomorphic to G? (b) Given a permutation group G acting on n letters, can one construct a graph of n vertices whose group of automorphisms is G? The former has been answered affirmatively by Frucht in [4] and [5], and many others. Concerning the latter, Kagno in [6] investigated the graphs of vertices ^ 6 and their group of automorphisms. It is known that not every group can have a graph in the sense of (b). For instance, letting