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Diameter and Girth of Representation Graphs of Quadratic Forms

Nico Lorenz, Marc Christian Zimmermann

TL;DR

This work studies the diameter and girth of representation graphs $\mathcal{G}_{q,a}$ built from non-degenerate quadratic forms over fields, with a focus on finite fields. It develops a unified, algebraic approach based on Witt decompositions and the action of the isometry group $\mathrm{iso}(q)$ to reduce path and cycle questions to low-dimensional subforms, enabling explicit diameter results and cycle counts. Over finite fields, the graphs are mostly connected with concrete diameters, except for a few small-field or anisotropic cases, and the minimum cycle length (girth) is largely 3 or 4 depending on subform structure and characteristic; the authors also provide detailed counts for 3- and 4-cycles, including exact formulas in many cases and a noteworthy exception for $\mathbb{H}\perp[1,1]$ over $\mathbb{F}_2$. Overall, the paper links quadratic-form invariants to precise combinatorial graph invariants, offering exact finite-field enumerations and insights that extend to certain infinite-field settings via structural reductions.

Abstract

Let $q$ be a non-degenerate quadratic form defined on an $F$ vector space $V$ and $a \in F$. We consider the Cayley graph on $V$ with generating set $\{x \in V \mid q(x) = a\}$ and study its diameter and girth. In particular, if $F$ is a finite field, we calculate these invariants and the number of cycles of minimal length in these graphs.

Diameter and Girth of Representation Graphs of Quadratic Forms

TL;DR

This work studies the diameter and girth of representation graphs built from non-degenerate quadratic forms over fields, with a focus on finite fields. It develops a unified, algebraic approach based on Witt decompositions and the action of the isometry group to reduce path and cycle questions to low-dimensional subforms, enabling explicit diameter results and cycle counts. Over finite fields, the graphs are mostly connected with concrete diameters, except for a few small-field or anisotropic cases, and the minimum cycle length (girth) is largely 3 or 4 depending on subform structure and characteristic; the authors also provide detailed counts for 3- and 4-cycles, including exact formulas in many cases and a noteworthy exception for over . Overall, the paper links quadratic-form invariants to precise combinatorial graph invariants, offering exact finite-field enumerations and insights that extend to certain infinite-field settings via structural reductions.

Abstract

Let be a non-degenerate quadratic form defined on an vector space and . We consider the Cayley graph on with generating set and study its diameter and girth. In particular, if is a finite field, we calculate these invariants and the number of cycles of minimal length in these graphs.

Paper Structure

This paper contains 16 sections, 27 theorems, 66 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quadratic Forms
  4. Graph Theory
  5. Connectedness and diameter
  6. First Examples
  7. Finite fields
  8. Structural properties of the diameter
  9. Representation graphs for $a=0$
  10. Hyperbolic planes
  11. On paths of length two
  12. Finishing the proof
  13. Girth
  14. Determining the Girth
  15. Cycles of Length 3
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $F$ be a finite field, $n \in \mathbb{N}$ a positive integer and $k = \lfloor \frac{n}{2} \rfloor$.

Figures (1)

  • Figure 1: The connected component of $(0,0)$ in the representation graph $\mathcal{G}_{\mathbb{H},1}$ of the hyperbolic plane $\mathbb{H}$ over (a): $\mathbb{F}_2$ and (b): $\mathbb{F}_4$.

Theorems & Definitions (55)

  • Theorem 2.1: book:Scharlau
  • Remark 2.2
  • Proposition 2.3: book:lidl
  • Corollary 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6
  • Theorem 3.1
  • proof
  • Conjecture 3.2
  • ...and 45 more