Its parameters as a strongly regular graph are (729,112,1,20).
This means that it has 729 vertices, and 40824 edges (112 per vertex).
Each edge is in a unique triangle (it is a locally linear graph) and each non-adjacent pair of vertices have exactly 20 shared neighbors.
[2] The construction of this graph involves the 56-point cap set in
This is a subset of points with no three in line in the five-dimensional projective geometry over a three-element field, and is unique up to symmetry.
, can be partitioned into a six-dimensional affine space
, which forms the set of points at infinity with respect to the affine space.
The Games graph has as its vertices the 729 points of the affine space
[1] Several of the graph's properties follow immediately from this construction.
vertices, because the number of points in an affine space is the size of the base field to the power of the dimension.
, and the three cap set points of the three lines would all lie on the intersection of this plane with
But this would violate the defining property of a cap set that it has no three points on a line, so no such extra triangle can exist.
The remaining property of strongly regular graphs, that all non-adjacent pairs of points have the same number of shared neighbors, depends on the specific properties of the 5-dimensional cap set.
, producing a smaller strongly regular graph with parameters (243,22,1,2).