Polynomial graph invariants from homomorphism numbers

TL;DR

This paper introduces a method to generate strongly polynomial graph sequences based on homomorphism counts, unifying several known graph polynomials and relating graph automorphisms to polynomial properties.

Contribution

It presents a new construction for strongly polynomial graph sequences, includes multiple classical polynomials, and introduces the branching core size parameter linking automorphisms to polynomial families.

Findings

The method generates families including the Tutte and chromatic polynomials.

Graphs with bounded branching core size are contained in finitely many polynomial sequences.

The approach unifies various graph invariants under a common polynomial framework.

Abstract

We give a method of generating strongly polynomial sequences of graphs, i.e., sequences $(H_{\mathbf{k}})$ indexed by a multivariate parameter $\mathbf{k}=(k_1,\ldots, k_h)$ such that, for each fixed graph $G$, there is a multivariate polynomial $p(G;x_1,\ldots, x_h)$ such that the number of homomorphisms from $G$ to $H_{\mathbf{k}}$ is given by the evaluation $p(G;k_1,\ldots, k_h)$. A classical example is the sequence $(K_k)$ of complete graphs, for which ${\rm hom}(G,K_k)=P(G;k)$ is the evaluation of the chromatic polynomial at $k$. Our construction produces a large family of graph polynomials that includes the Tutte polynomial, the Averbouch-Godlin-Makowsky polynomial and the Tittmann-Averbouch-Makowsky polynomial. We also introduce a new graph parameter, the {\em branching core size} of a simple graph, related to how many involutive automorphisms with fixed points it has. We prove that a countable family of graphs of bounded branching core size (which in particular implies bounded tree-depth) is always contained in a finite union of strongly polynomial sequences.