Tutte Algebras of Graphs and Formal Group Theory

In 1947, in a beautiful and influential paper [15], Tutte introduced a powerful algebraic framework for constructing invariants of multigraphs. This has spread increasing ripples down the years, and is, for example, a forerunner to certain recent exciting developments in knot theory [5]. However, combinatorialists have tended to play down the algebraic components of Tutte's work, and concentrate instead on the graph theory. Indeed, in the process of improving and extending his original ideas, Tutte himself [16] has felt the need to present the material 'cleared of algebraic superstructure'. It is our contention that this superstructure is actually remarkably interesting, and that part of it exposes links between combinatorics and the theory of formal groups, and thereby also with the algebraic topology of spaces such as loops on the 3-sphere and infinite-dimensional complex projective space [10]. Our aim here is to concentrate on one aspect of Tutte's ring, which is polynomial in infinitely many variables over Z, by restricting attention to simple graphs. This reduces the number of variables to one, but has the attraction that we may enrich the structure in a natural and appealing way to that of a Hopf algebra supporting a difference operator (actually the discrete derivative); in this procedure, a graph becomes identified with its chromatic polynomial. These topics are the subject matter of our § 1 and § 2. We then generalise so as to include recent developments in umbral calculus [8], which acts as the bridge with the theory of formal groups. This involves incorporating into the algebraic construction information concerning partitions of the vertices of the graphs, as explained in § 3 and § 4. There again results a polynomial algebra in one variable, but this time over a larger ring <£*. Once more we can naturally impose the structure of a Hopf algebra, which now supports an umbral difference operator, and a graph becomes identified with its umbral chromatic polynomial, as introduced in [12]. We are thus able in § 5 to establish a combinatorial interpretation for an arbitrary formal group over any ring free of additive torsion. This of course includes the universal case, and raises several interesting questions. We save a discussion of the interactions with algebraic topology for the future. We refer the reader to Aigner [1] for general combinatorial terminology, to S weedier [14] for all information concerning Hopf algebras, and to Hazewinkel [4] for an encyclopaedic description of the theory of formal groups. It is a pleasure to acknowledge enjoyable and fruitful discussions with several mathematicians during the development of this material. In particular, Bill Schmitt provided privileged access to his ongoing work on combinatorial Hopf algebras, and so helped inspire § 2; we are currently working together on a sequel