Binary invariants and orientations of graphs

Abstract

It is well known that Julius Petersen’s famous paper on graph factorisation [9] has its origin in a problem which arose in connection with Hilbert’s proof of the Finite Basis Theorem for the invariants of binary forms [3]. Once one accepts the graph theoretic framework, the reformulation of the algebraic problem is straightforward and needs only a minimum of explanation-at least in the context of a hundred years ago-and this is precisely what Petersen provides in his paper. It is, however, unfortunate that he chose not to give any indication of the motivation for the fundamental idea of using graphs in connection with invariants, except in a very oblique way by acknowledging that Sylvester had also worked on the problem and that there had been some correspondence between them ([9, p. 1941, [13]). Indeed, Petersen’s whole approach is based on an observation made by Sylvester in 187~ignored by most invariant theorists at the time-to the effect that regular graphs contain essentially the same information as invariants of binary forms, in the sense that there is a natural map which assigns to each regular graph of order n an invariant of the binary form of order n, and that the invariants so obtained additively generate all invariants [14]. Instead of the unwieldy invariants one can therefore study the intuitively much more accessible graphs. In 1891 this was by no means self-evident, and Petersen’s silence on this point may well have contributed to the fact that his paper remained an isolated occurrence. In the present paper we shall take as our point of departure a problem which lies at the heart of the correspondence between graphs and invariants, and to which Petersen had given a certain amount of thought. The map graphs ---, invariants is many-to-one; it is therefore of interest to know which graphs lie in its ‘kernel’, i.e., give rise to the zero invariant. Petersen recognised the importance