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Games graph

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Source updated June 23, 2026

In graph theory, the Games graph is the largest known locally linear strongly regular graph. 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. It is named after Richard A. Games, who suggested its construction in an unpublished communication and wrote about related constructions.

01Construction

The construction of this graph involves the 56-point cap set in $PG(5,3)$ . 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. The six-dimensional projective geometry, $PG(6,3)$ , can be partitioned into a six-dimensional affine space $AG(6,3)$ and a copy of $PG(5,3)$ , 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 $AG(6,3)$ . Each line in the affine space goes through three of these points, and through a fourth point at infinity. The graph contains a triangle for every line of three affine points that passes through a point of the cap set.

02Properties

Several of the graph's properties follow immediately from this construction. It has $729=3^{6}$ vertices, because the number of points in an affine space is the size of the base field to the power of the dimension. For each affine point, there are 56 lines through cap set points, 56 triangles containing the corresponding vertex, and $112=56\times 2$ neighbors of the vertex. And there can be no triangles other than the ones coming from the construction, because any other triangle would have to come from three different lines meeting in a common plane of $PG(6,3)$ , and the three cap set points of the three lines would all lie on the intersection of this plane with $PG(5,3)$ , which is a line. 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.

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Sources and credits

This article is adapted from the Wikipedia article “Games graph”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Games graph

01Construction

02Properties

03Related graphs

Sources and credits