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# The affine polar graphs on 64 vertices

We consider two affine polar graphs on 64 vertices,
namely VO^{+}(6,2) and VO^{−}(6,2).
## VO^{+}(6,2)

The graph VO^{+}(6,2) is strongly regular with parameters
v = 64, k = 35, λ = 18, μ = 20 and spectrum
35^{1} 3^{35} (−5)^{28}.
It is isomorphic to the quadratic forms graph on GF(2)^{3},
and to the alternating forms graph on GF(2)^{4},
and to the complement of the folded halved 8-cube.

The local graphs (induced on the neighbours of a vertex)
are orthogonal graphs O^{+}(6,2), strongly regular
with parameters v = 35, k = 18, λ = μ = 9 and spectrum
18^{1} 3^{14} (−3)^{20}. These are isomorphic to
the graph on the lines of PG(3,2), adjacent when intersecting.

The graphs induced on the nonneighbours of a vertex
are (non)orthogonal graphs NO^{+}(6,2), strongly regular
with parameters v = 28, k = 15, λ = 6, μ = 10 and spectrum
15^{1} 1^{20} (−5)^{7}. These are isomorphic to the
complement of the triangular graph T(8).

## VO^{−}(6,2)

The graph VO^{−}(6,2) is strongly regular with parameters
v = 64, k = 27, λ = 10, μ = 12 and spectrum
27^{1} 3^{36} (−5)^{27}.
The local graphs are orthogonal graphs O^{−}(6,2),
strongly regular with parameters v = 27, k = 10, λ = 1, μ = 5
and spectrum 10^{1} 1^{20} (−5)^{6}.
These are isomorphic to the collinearity graph of GQ(2,4),
and to the complement of the Schläfli graph.

The graphs induced on the nonneighbours of a vertex
are (non)orthogonal graphs NO^{−}(6,2), strongly regular
with parameters v = 36, k = 15, λ = μ = 6 and spectrum
15^{1} 3^{15} (−3)^{20}.

## Regular 2-graphs

These affine polar graphs and nonorthogonal graphs are
graphs in the switching class of regular 2-graphs with
2-transitive groups Sp(8,2) and Sp(6,2), respectively.
Switching a point isolated yields strongly regular graphs
on 63, 35, or 27 vertices. The parameters are
(i) v = 63, k = 30, λ = 13, μ = 15 and spectrum
30^{1} 3^{35} (−5)^{27}.
This is the orthogonal graph O(7,2), isomorphic to the
symplectic graph Sp(6,2).

(ii) v = 35, k = 18, λ = μ = 9.
This is the orthogonal graph O^{+}(6,2) already seen above.

(iii) v = 27, k = 10, λ = 1, μ = 5.
This is the orthogonal graph O^{−}(6,2) already seen above.