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# Games Graph

The Games graph is a strongly regular graph on 729 vertices with parameters .

It is distance-regular but not distance-transitive with intersection array and has graph spectrum .

It is implemented in the Wolfram Language as GraphData["GamesGraph"].

It can be constructed as follows. There is a unique 56-cap in (i.e., the set of 56 points such that any line meets it in at most two points) (Hill 1978). Taking the as vertices the points of and joining two vertices when the line through the points meets the hyperplane at infinity in a point of the cap gives the Games graph (Cameron 1975, Games pers. comm. to Brouwer and van Lint 1984).

Distance-Regular Graph, Strongly Regular Graph

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## References

Bondarenko, A. V. and Radchenko, D. V. "On a Family of Strongly Regular Graphs with ." J. Combin. Th. (B) 103, 521-531, 2013.Brouwer, A. E. and van Lint, J. H. "Strongly Regular Graphs and Partial Geometries." In Enumeration and Design: Papers from the Conference on Combinatorics Held at the University of Waterloo, Waterloo, Ont., June 14-July 2, 1982 (Ed. D. M. Jackson and S. A. Vanstone). Toronto, Canada: Academic Press, pp. 85-122, 1984.Brouwer, A. E. and van Maldeghem, H. "The Games Graph." §10.75 in Strongly Regular Graphs. Cambridge, England: Cambridge University Press, p. 354-355, 2022.Cameron, P. J. "Partial Quadrangles." Quart. J. Math. Oxford 26, 61-73, 1975.DistanceRegular.org. "Games graph." https://www.distanceregular.org/graphs/games.html.Games, R. A. "The Packing Problem for Finite Projective Geomeries." Ph.D. Thesis. Columbus, OH: Ohio State Univ., pp. 171 and 329, 1980.Hill, R. "Caps and Codes." Discr. Math. 22, 111-137, 1978.

Games Graph

## Cite this as:

Weisstein, Eric W. "Games Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GamesGraph.html