Ask zeta functions of joins of graphs

TL;DR

This paper investigates how ask zeta functions, which encode kernel sizes of matrices related to graphs, behave under the graph operation of joining, with implications for counting conjugacy classes in graphical groups.

Contribution

It demonstrates that two specific ask zeta functions are well-behaved under graph joins, extending previous work on their properties and applications.

Findings

Ask zeta functions are preserved under graph joins.

Results apply to enumeration of conjugacy classes in graphical groups.

Provides new tools for analyzing algebraic structures via graph operations.

Abstract

In previous work (arXiv:1908.09589), we studied rational generating functions ("ask zeta functions") associated with graphs and hypergraphs. These functions encode average sizes of kernels of generic matrices with support constraints determined by the graph or hypergraph in question, with applications to the enumeration of linear orbits and conjugacy classes of unipotent groups. In the present article, we turn to the effect of a natural graph-theoretic operation on associated ask zeta functions. Specifically, we show that two instances of rational functions, $W^-_\Gamma(X,T)$ and $W^\sharp_\Gamma(X,T)$, associated with a graph $\Gamma$ are both well-behaved under taking joins of graphs. In the former case, this has applications to zeta functions enumerating conjugacy classes associated with so-called graphical groups.