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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 351 335 (−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 181 314 (−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 151 120 (−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 271 336 (−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 101 120 (−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 151 315 (−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 301 335 (−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.