Transition polynomials

Abstract

Transition polynomials for 4-regular graphs were defined by Jaeger [On transition polynomials of 4-regular graphs, in: Hahn et al. (Eds.), Cycles and Rays, Kluwer Academic Publishers, Dordrecht, 1990, pp. 123–150.]. Many polynomials with a wide range of applications in mathematics and physics are transition polynomials. Such is the case for Penrose, Martin and Kauffman bracket polynomials. The Tutte polynomial of a plane graph is also a transition polynomial (in the case x = y ). The authors in [J.A. Ellis-Monaghan, I. Sarmiento, Medical graphs and the Penrose polynomial, Congr. Numer. 150 (2001) 211–222.] generalized Jaeger's transition polynomials to a class of graphs including planar drawings of non-planar graphs. A further generalization was obtained in [J.A. Ellis-Monaghan, I. Sarmiento, Generalized transition polynomials, Congr. Numer. 150 (2003) 211–222.] where the authors defined transition polynomials for all Eulerian graphs. In [J.A. Ellis-Monaghan, I. Sarmiento, Medical graphs and the Penrose polynomial, Congr. Numer. 150 (2001) 211–222.] and [J.A. Ellis-Monaghan, I. Sarmiento, Generalized transition polynomials, Congr. Numer. 150 (2003) 211–222.] transition polynomials were seen as homomorphisms of Hopf algebras. Following a different approach, Aigner and Mielke [The Penrose polynomial of binary matroids, Monatsh. Math. 131 (2000) 1–13.] defined a transition polynomial for isotropic systems associated to binary matroids. In this paper we will present an overview of the very rich world of transition polynomials.