Diameter of hyperbolic-plane representation graphs over finite fields

Prove that for every finite field F with f elements and every a ∈ F*, the representation graph G_{H,a} of the hyperbolic plane H satisfies diam(G_{H,a}) = 4 when f ≤ 9 and diam(G_{H,a}) = 3 when f ≥ 11.

Background

The paper studies representation graphs G_{q,a} whose vertices are the vectors of a quadratic-form space and whose edges join vertices whose difference has quadratic-form value a. For the hyperbolic plane H over a finite field F and a nonzero parameter a, the authors establish connectedness when |F| ≥ 5 and prove the general bounds 3 ≤ diam(G_{H,a}) ≤ 4.

Computer algebra computations for finite fields with up to 2000 elements indicate that diameter 4 occurs only for very small fields and that diameter 3 occurs once the field has at least 11 elements. The authors explicitly state that they were unable to prove a uniform threshold result and therefore formulate the precise finite-field diameter pattern as a conjecture.

References

We used a computer algebra system to explicitly compute the diameter of representation graphs of the hyperbolic plane over finite fields with up to 2000 elements. These computations suggest that the diameter of a representation graph of a hyperbolic plane is $4$ only for very small fields and is $3$ once the field has at least $11$ elements. Unfortunately, we were unable to find any integer $f$ for which we could prove that $diam(G_{H,a}) = 3$ for all finite fields with $|F| \geq f$. So we offer only the following conjecture.

Let $F$ be a finite field with $f$ elements. For $a \in F\ast$ we have $$diam(G_{H, a}) = \begin{cases} 4, &\text{ if $f \leq 9$}\ 3, &\text{ if $f \geq 11$} \end{cases}$$

— Diameter and Girth of Representation Graphs of Quadratic Forms  (2503.01721 - Lorenz et al., 3 Mar 2025) in Section 3, subsection “Hyperbolic planes,” immediately before the Conjecture environment