The graphs VO^{+}(2*e*,*q*) are strongly regular,
with parameters
*v* = *q*^{2e},
*k* = (*q*^{e−1} + 1)(*q*^{e} − 1),
λ = *q*(*q*^{e−2} + 1)(*q*^{e−1} − 1)
+ *q* − 2,
μ = *q*^{e−1}(*q*^{e−1} + 1),
*r* = *q*^{e} − *q*^{e−1} − 1,
*s* = − *q*^{e−1} − 1.

The graphs VO^{−}(2*e*,*q*) are strongly regular,
with parameters
*v* = *q*^{2e},
*k* = (*q*^{e−1} − 1)(*q*^{e} + 1),
λ = *q*(*q*^{e−2} − 1)(*q*^{e−1} + 1)
+ *q* − 2,
μ = *q*^{e−1}(*q*^{e−1} − 1),
*r* = *q*^{e−1} − 1,
*s* = − *q*^{e} + *q*^{e−1} − 1.

The graphs VO(2*e*+1,*q*) are not strongly regular.

For details about the local structure of these graphs, see

A.E. Brouwer & E.E. Shult, *Graphs with odd cocliques*,
Eur. J. Combin. **11** (1990) 99-104.

For *d* = 2 we find that VO^{−}(2,*q*) is a coclique, and
that VO^{+}(2,*q*) is a *q*×*q* grid graph
(that is, H(2,*q*)).

Note that the graphs VO^{+}(2*e*,2) and
VO^{−}(2*e*,2) have λ = μ − 2
and hence give rise to SBIBDs with parameters
2-(2^{2e}, 2^{2e−1} + 2^{e−1}−1,
2^{2e−2}+ 2^{e−1}) and
2-(2^{2e}, 2^{2e−1} − 2^{e−1}−1,
2^{2e−2} − 2^{e−1}), respectively.

If we take the Hamming scheme H(*n*,4) and call two vertices adjacent
if their distance is even we obtain a strongly regular graph
(as was observed in
S. Kageyama, G.M. Saha & A.D. Das,
*Reduction of the number of association classes of hypercubic
association schemes*, Ann. Inst. Stat. Math. **30** (1978) 115-123).
But this is just VO^{+}(2*n*,2) or VO^{−}(2*n*,2)
(where the sign is (−1)^{n}). Indeed, the weight of a quaternary
digit is given by the (elliptic) binary quadratic form
*x*_{1}^{2}
+ x_{1}x_{2} + x_{2}^{2}.