TWO-GRAPHS, A SECOND SURVEY

Publisher Summary A two-graph ( Ω , Δ ) is a vertex set Ω and a collection Δ of 3-subsets of Ω such that every 4-subset of Ω contains an even number of 3-subsets from Δ . The two-graph is regular if every 2-subset of Ω is contained in the same number of 3-subsets from Δ . The chapter describes two-graphs in terms of exterior algebra and presents Cameron's simple proof as per which two-graphs and Euler graphs are equal in number. It also presents the relations with equiangular lines, and highlights certain generalizations and Cameron's cohomology classes associated with a group of automorphisms of a two-graph by a number of simple examples. The example with nontrivial first and second invariant is completely worked out. The chapter also explains the state of affairs for regular two-graphs for n ≤ 46 and the construction of conference two-graphs of order pq 2 + 1 .

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