Add construction of strongly regular digraph

src/doc/en/reference/references/index.rst

@@ -2174,6 +2174,11 @@ REFERENCES:
 .. [Duv1983] J.-P. Duval, Factorizing words over an ordered alphabet,
  J. Algorithms 4 (1983) 363--381.
 
+.. [Duv1988] \A. Duval.
+ *A directed graph version of strongly regular graphs*,
+ Journal of Combinatorial Theory, Series A 47(1) (1988): 71-100.
+ :doi:`10.1016/0097-3165(88)90043-X`
+
 .. [DW1995] Andreas W.M. Dress and Walter Wenzel, *A Simple Proof of
  an Identity Concerning Pfaffians of Skew Symmetric
  Matrices*, Advances in Mathematics, volume 112, Issue 1,

src/sage/graphs/digraph_generators.py

@@ -349,6 +349,53 @@ def Path(self, n):
  g.set_pos({i: (i, 0) for i in range(n)})
  return g
 
+ def StronglyRegular(self, n):
+ r"""
+ Return a Strongly Regular digraph with `n` vertices.
+
+ The adjacency matrix of the graph is constructed from a skew Hadamard
+ matrix of order `n+1`. These graphs were first constructed in [Duv1988]_.
+
+ INPUT:
+
+ - ``n`` -- integer, the number of vertices of the digraph.
+
+ .. SEEALSO::
+
+ - :func:`sage.combinat.matrices.hadamard_matrix.skew_hadamard_matrix`
+ - :meth:`Paley`
+
+ EXAMPLES:
+
+ A Strongly Regular digraph satisfies the condition `AJ = JA = kJ` where
+ `A` is the adjacency matrix::
+
+ sage: g = digraphs.StronglyRegular(7); g
+ Strongly regular digraph: Digraph on 7 vertices
+ sage: A = g.adjacency_matrix()*ones_matrix(7); B = ones_matrix(7)*g.adjacency_matrix()
+ sage: A == B == A[0, 0]*ones_matrix(7)
+ True
+
+ TESTS:
+
+ Wrong parameter::
+
+ sage: digraphs.StronglyRegular(73)
+ Traceback (most recent call last):
+ ...
+ ValueError: strongly regular digraph with 73 vertices not yet implemented
+
+ """
+ from sage.combinat.matrices.hadamard_matrix import skew_hadamard_matrix
+ from sage.matrix.constructor import ones_matrix, identity_matrix
+ if skew_hadamard_matrix(n+1, existence=True) is not True:
+ raise ValueError(f'strongly regular digraph with {n} vertices not yet implemented')
+
+ H = skew_hadamard_matrix(n+1, skew_normalize=True)
+ M = H[1:, 1:]
+ M = (M + ones_matrix(n)) / 2 - identity_matrix(n)
+ return DiGraph(M, format='adjacency_matrix', name=f'Strongly regular digraph')
+
  def Paley(self, q):
  r"""
  Return a Paley digraph on `q` vertices.