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Preprint Number 21

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21. J.A. Makowsky and B. Zilber
Polynomial invariants of graphs and totally categorical theories
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Submission date: 13 December 2006

Abstract:

In the analysis of the structure of totally categorical first order theories, the second author showed that certain combinatorial counting functions play an important role. Those functions are {\em invariants of the structures} and are always polynomials in one or many variables, depending on the number of independent dimensions of the theory in question.

The first author introduced the notion of graph polynomials definable in Monadic Second Order Logic, and showed that the Tutte polynomial and its generalization, the matching polynomial, the cover polynomial and the various interlace polynomials fall into this category. This definition can be extended to allow definability in full second order, or even higher order Logic.

The purpose of this paper is to show that many graph polynomials and combinatorial counting functions of graph theory do occur as combinatorial counting functions of totally categorical theory. We also give a characterization of polynomials definable in Second Order Logic.

Mathematics Subject Classification: 03C, 03C98, 05A, 05A15, 05C, 05C15.

Keywords and phrases: Combinatorial counting, categoricity, graph polynomials

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