A q-analogue of graph independence polynomials with a group-theoretic interpretation

TL;DR

This paper introduces a q-analogue of graph independence polynomials using totally-isotropic polynomials of alternating matrix spaces, linking algebraic structures with combinatorial graph invariants over finite fields.

Contribution

It defines totally-isotropic polynomials for alternating matrix spaces and shows they form a natural q-analogue of graph independence polynomials, with applications to p-groups and graphical groups.

Findings

Totally-isotropic polynomials generalize independence polynomials.

The polynomials enumerate abelian subgroups of p-groups.

Implications for graphical groups over finite fields.

Abstract

We define totally-isotropic polynomials of alternating matrix spaces over finite fields, by analogy with independence polynomials of graphs. Our main result shows that totally-isotropic polynomials of graphical alternating matrix spaces give rise to a natural q-analogue of graph independence polynomials. For p-groups of class 2 and exponent p, this family of polynomials over fields of order p can be naturally interpreted as enumerating their abelian subgroups containing the commutator subgroup according to the orders. With this interpretation, our main result has implications to graphical groups over finite fields, in the same spirit as the results in (Bull. Lond. Math. Soc., 2022) by Rossmann, who studied enumerating conjugacy classes of graphical groups over finite fields.