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

Abstract

We define totally‐isotropic space polynomials of alternating matrix spaces over finite fields, by analogy with independence polynomials of graphs. Our main result shows that totally‐isotropic space polynomials of graphical alternating matrix spaces give rise to a natural ‐analogue of graph independence polynomials. For ‐groups of class 2 and exponent , this family of polynomials over fields of order 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 where is an odd prime, 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.