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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 $d(x,y)=a$.

Proposition 4 (p. 235, "Some graph isomorphisms"). For fixed $q$ and $n$, all the graphs $E_q(n,a)$ with $a$ a nonzero square, $a=b^2$ for some $b\ne0$, are isomorphic to one another, and all those with $a$ a nonsquare are isomorphic to one another. Hence the graphs with $a\ne0$ form at most two isomorphism classes.

With $E_q(n,0)$ this gives at most three nonisomorphic graphs $E_q(n,a)$ for each $\mathbb F_q^n$, as the paper states on pp. 224 and 235.

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: Proposition 4 on p. 235, its proof on pp. 235-236, the count of classes on pp. 224 and 235. The edition read is identified on the source card.

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

Proof pointer

Pp. 235-236. For $c\ne0$ the map $y\mapsto cy$ multiplies every distance by $c^2$, so it carries $E_q(n,a)$ onto $E_q(n,c^2a)$.