R of Ways to Label a Structure

in accordance with the theorem in Katz [1955]. This result was anticipated in turn by Rubin and Sitgreaves in an unpublished memorandum cited in Katz [1955]. In order to contribute something positive in this note, we now prove the theorem about graphs and groups which justifies the formula given in Harary and Read [1966] for the number of ways to label a structure. Since this is a sequel to Harary and Read [1966], its notation and terminology will be used. Thus we write s(G) for the symmetry number of graph G (the order of its automorphism group r(G)) and l(G) for the number of labelings of G. As usual we denote the number of points of G by p. The notation used in the following proof follows that in Harary [in press] and Harary and Palmer [1965]. Accordingly, S~ is the symmetric group of degree p acting on X = {1, 2, . . , p}; X (~) is the set of unordered pairs of the objects in X; S~ (~) is the pair group acting on X (~) as induced by S~ ; and E2 is the identity group on Y = {0, !}" The power group (introduced in Harary and Palmer [1965]) E~ ~'~ acts on yxc. , and each function ] from X ~2) into Y represents a labeled graph with point set X. Two points i, j ~ X are considered adjacent in the graph of ] whenever ]({i, j}) = I.