Equiangular line systems and switching classes containing regular graphs

Abstract We develop the theory of equiangular lines in Euclidean spaces. Our focus is on the question of when a Seidel matrix having precisely three distinct eigenvalues has a regular graph in its switching class. We make some progress towards an answer to this question by finding some necessary conditions and some sufficient conditions. Furthermore, we show that the cardinality of an equiangular line system in 18 dimensional Euclidean space is at most 60.

[1]  Willem H. Haemers,et al.  A (49, 16, 3, 6) Strongly Regular Graph Does Not Exist , 1989, Eur. J. Comb..

[2]  Akihiro Munemasa,et al.  Equiangular lines in Euclidean spaces , 2014, J. Comb. Theory, Ser. A.

[3]  Boris Bukh,et al.  Bounds on Equiangular Lines and on Related Spherical Codes , 2015, SIAM J. Discret. Math..

[4]  Wei-Hsuan Yu,et al.  Upper bounds for $s$-distance sets and equiangular lines , 2016, 1611.09479.

[5]  J. Seidel.  Graphs and two-graphs , 1974 .

[6]  Dominique de Caen,et al.  Large Equiangular Sets of Lines in Euclidean Space , 2000, Electron. J. Comb..

[7]  Willem H. Haemers,et al.  Spectra of Graphs , 2011 .

[8]  Patric R. J. Östergård,et al.  Enumeration of Seidel matrices , 2017, Eur. J. Comb..

[9]  J. J. Seidel,et al.  Equilateral point sets in elliptic geometry , 1966 .

[10]  Tilen Marc,et al.  There is no (75,32,10,16) strongly regular graph , 2015, Linear Algebra and its Applications.

[11]  Tero Harju,et al.  Euler Graphs, Triangle-Free Graphs and Bipartite Graphs in Switching Classes , 2003, Fundam. Informaticae.

[12]  Ferenc SzollHosi.  A remark on a construction of D.S. Asche , 2017, 1703.04505.

[13]  Tilen Marc,et al.  There is no (95, 40, 12, 20) strongly regular graph , 2016, Journal of Combinatorial Designs.

[14]  D. E. Taylor.  Regular 2‐Graphs , 1977 .

[15]  Alexander Barg,et al.  New bounds for equiangular lines , 2013, Discrete Geometry and Algebraic Combinatorics.

[16]  Ferenc Szöllősi.  A Remark on a Construction of D.S. Asche , 2017 .