A logician's view of graph polynomials

TL;DR

This paper explores graph polynomials through a model theoretic lens, distinguishing their semantic and syntactic aspects, and introduces a logical framework for their representation and comparison.

Contribution

It provides a model theoretic perspective on graph polynomials, introduces a class definable by Second Order Logic, and clarifies the semantic versus algebraic properties of zeros and stability.

Findings

Graph polynomials can be characterized by their logical definability.

Location of zeros and stability are not semantic properties.

A unified framework for classical and new results in algebraic combinatorics.

Abstract

Graph polynomials are graph parameters invariant under graph isomorphisms which take values in a polynomial ring with a fixed finite number of indeterminates. We study graph polynomials from a model theoretic point of view. In this paper we distinguish between the graph theoretic (semantic) and the algebraic (syntactic) meaning of graph polynomials. We discuss how to represent and compare graph polynomials by their distinctive power. We introduce the class of graph polynomials definable using Second Order Logic which comprises virtually all examples of graph polynomials with a fixed finite set of indeterminates. Finally we show that the location of zeros and stability of graph polynomials is not a semantic property. The paper emphasizes a model theoretic view and gives a unified exposition of classical results in algebraic combinatorics together with new and some of our previously obtained results scattered in the graph theoretic literature.