Equiangular lines

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In geometry, a set of lines in Euclidean space is called equiangular if every pair of lines makes the same angle.

Equiangular lines are related to two-graphs. Given a set of equiangular lines, let c be the cosine of the common angle. We assume that angle is not 90°, since that case is trivial (i.e., not interesting, because the lines are just coordinate axes); thus, c is nonzero. We may move the lines so they all pass through the origin of coordinates. Choose one unit vector in each line. Form the matrix M of inner products. This matrix has 1 on the diagonal and ±c everywhere else, and it is symmetric. Subtracting the identity matrix I and dividing by c, we have a symmetric matrix with zero diagonal and ±1 off the diagonal. This is the adjacency matrix of a two-graph.

[edit] References

• van Lint, J. H., and Seidel, J. J. Equilateral point sets in elliptic geometry. Proc. Koninkl. Ned. Akad. Wetenschap. Ser. A 69 (= Indagationes Mathematicae 28) (1966), 335-348.
• Brouwer, A.E., Cohen, A.M., and Neumaier, A. Distance-Regular Graphs. Springer-Verlag, Berlin, 1989. Section 3.8.
• Godsil, Chris, and Royle, Gordon. Algebraic Graph Theory. Graduate Texts in Mathematics, Vol. 207. Springer-Verlag, New York, 2001. (See Chapter 11.)