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Ask zeta functions of joins of graphs

Tobias Rossmann, Christopher Voll

TL;DR

This paper advances the study of ask zeta functions associated with graphs and hypergraphs by analyzing how joins and reflexive-graph operations influence the rational functions $W^-_\Gamma(X,T)$, $W^+_\Gamma(X,T)$, and the new $W^\sharp_\Gamma(X,T)$. A central methodological innovation is the explicit combinatorial parameterisation of minors via selectors (hypergraphs) and animations (graphs), which yields a new Uniformity proof and allows sharp join formulae. The authors prove a Reflexive Graph Modelling Theorem, establishing that for reflexive graphs the three zeta-function families coincide with a modelling hypergraph’s zeta function, revealing rigidity phenomena. They further develop a dedicated framework for joins and reflexive joins, introduce the invariant $W^\sharp_\Gamma$, and connect these zeta-functions to graphical groups, enabling concrete calculations and explicit formulae for many graphs. Overall, the work provides both conceptual and computational tools to understand how graph operations shape ask zeta functions and related group-theoretic zeta data, with potential applications to class-counting in graphical group schemes and beyond.

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, $W^-_Γ(X,T)$ and $W^\sharp_Γ(X,T)$, associated with a graph $Γ$ 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.

Ask zeta functions of joins of graphs

TL;DR

This paper advances the study of ask zeta functions associated with graphs and hypergraphs by analyzing how joins and reflexive-graph operations influence the rational functions , , and the new . A central methodological innovation is the explicit combinatorial parameterisation of minors via selectors (hypergraphs) and animations (graphs), which yields a new Uniformity proof and allows sharp join formulae. The authors prove a Reflexive Graph Modelling Theorem, establishing that for reflexive graphs the three zeta-function families coincide with a modelling hypergraph’s zeta function, revealing rigidity phenomena. They further develop a dedicated framework for joins and reflexive joins, introduce the invariant , and connect these zeta-functions to graphical groups, enabling concrete calculations and explicit formulae for many graphs. Overall, the work provides both conceptual and computational tools to understand how graph operations shape ask zeta functions and related group-theoretic zeta data, with potential applications to class-counting in graphical group schemes and beyond.

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, and , associated with a graph 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.
Paper Structure (106 sections, 73 theorems, 106 equations, 1 table)

This paper contains 106 sections, 73 theorems, 106 equations, 1 table.

Table of Contents

  1. Introduction
  2. Ask zeta functions derived from matrices of linear forms
  3. From matrices of linear forms to ask zeta functions.
  4. Global templates for ask zeta functions and arithmetic questions.
  5. Matrices of linear forms from graphs and hypergraphs
  6. Matrices of linear forms from hypergraphs.
  7. Matrices of linear forms from graphs.
  8. Background: the Uniformity Theorem
  9. Result: joins of loopless graphs
  10. Enter $W^\flat_\Gamma$: renormalising $W^-_\Gamma$.
  11. Join powers.
  12. Related work: disjoint unions and Hadamard products
  13. Result: the Reflexive Graph Modelling Theorem
  14. A new modelling theorem.
  15. Rigidity of ask zeta functions.
  16. ...and 91 more sections

Key Result

Theorem 1.3

Theorems & Definitions (162)

  • Example 1.1
  • Example 1.2
  • Theorem 1.3: Uniformity Theorem cico
  • Remark 1.4
  • Theorem A: Loopless joins
  • Remark 1.5: Precursors to Theorem \ref{['thm:join']}
  • Remark 1.6
  • Corollary 1.7
  • proof
  • Example 1.8
  • ...and 152 more