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Statement

Notation as on the Theorem 1 page: $q$ is odd and $E_q(n,a)$ is the graph on $\mathbb F_q^n$ joining $x,y$ when $d(x,y)=a$.

Theorem 5 (p. 236, "More graph isomorphisms in even dimensions"). For even $n$ and fixed $q$, the graphs $E_q(n,a)$ are isomorphic for all nonzero $a$. So for each $\mathbb F_q^n$ with $n$ even there are exactly two nonisomorphic graphs, $E_q(n,0)$ and $E_q(n,1)$.

This sharpens Proposition 4, which allows two classes for nonzero $a$ in every dimension. The paper contrasts the count with the finite upper half plane graphs, where it reports that $q$ distinct graphs appear to arise for each $\mathbb F_q$, a claim it says remains to be proved (p. 235). In odd dimension the paper gives only the upper bound of three classes.

Source. A. Medrano, P. Myers, H. M. Stark and A. Terras, Finite analogues of Euclidean space, J. Comput. Appl. Math. 68 (1996), 221-238, doi:10.1016/0377-0427(95)00261-8: the discussion on p. 235, Theorem 5 and its proof on p. 236. The edition read is identified on the source card.

Read depth. Claims checked: the statement and the proof were read on the printed pages. Nothing here is independently reviewed.

Proof pointer

P. 236. Every $c\ne0$ is a sum of two squares, $c=x_1^2+x_2^2$. The matrix with $k$ copies of the block $\begin{pmatrix}x_1&-x_2\\x_2&x_1\end{pmatrix}$ down the diagonal, $n=2k$, multiplies each distance by $c$, by the two-square identity $(x_1y_1-x_2y_2)^2+(x_2y_1+x_1y_2)^2=(x_1^2+x_2^2)(y_1^2+y_2^2)$ applied to each pair of coordinates. It therefore maps $E_q(n,a)$ onto $E_q(n,ca)$. The graphs $E_q(n,0)$ and $E_q(n,1)$ are distinguished by their degrees from Theorem 1.