The interlace polynomial of a graph

  • Authors:

  • Richard Arratia;Béla Bollobás;Gregory B. Sorkin

  • Affiliations:

  • Department of Mathematics, University of Southern California, Los Angeles, CA;Department of Mathematical Sciences, University of Memphis, Memphis, TN and Trinity College, Cambridge CB2 1TQ, UK;Department of Mathematical Sciences, IBM T.J. Watson Research Center, Yorktown Heights, NY

  • Venue:
  • Journal of Combinatorial Theory Series B - Special issue dedicated to professor W. T. Tutte
  • Year:
  • 2004

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Abstract

Motivated by circle graphs, and the enumeration of Euler circuits, we define a one-variable "interlace polynomial" for any graph. The polynomial satisfies a beautiful and unexpected reduction relation, quite different from the cut and fuse reduction characterizing the Tutte polynomial.It emerges that the interlace graph polynomial may be viewed as a special case of the Martin polynomial of an isotropic system, which underlies its connections with the circuit partition polynomial and the Kauffman brackets of a link diagram. The graph polynomial, in addition to being perhaps more broadly accessible than the Martin polynomial for isotropic systems, also has a two-variable generalization that is unknown for the Martin polynomial. We consider extremal properties of the interlace polynomial, its values for various special graphs, and evaluations which relate to basic graph properties such as the component and independence numbers.