experiments/flower-snarks: prove that 16 is the largest eigenvalue

experiments/flower-snarks.py

@@ -15,12 +15,49 @@ sys.path.append(os.path.dirname(__file__) + "/..")
 from graph_tools.all import *
 from sage.all import identity_matrix, QQ, DiGraph, matrix
 
+
+def deflate_eigenvector(M, u):
+  coord = min(( r for r in range(u.nrows()) if u[r, 0] > 0 ), key=lambda r: u[r, 0])
+  a = M[coord, :]
+  return M - u * a / u[coord, 0]
+
+
+def bound_spectral_radius(M, bound):
+  def norm(A):
+    """The matrix 1-norm. Max of the sum of columns.
+
+    We do not use the norms from Sage because they are slow
+    for sprase matrices.
+    """
+    sums = [ 0 ] * A.ncols()
+    for (r, c), v in A.dict().items():
+      sums[c] += abs(v)
+
+    return max(sums)
+
+  X = M
+  p = 1
+
+  while True:
+    b = bound**p
+    n = norm(X)
+    print("  p=%i upper bound is %0.3lg" % (p, n**(1/p)))
+
+    if n <= b: break
+
+    X = X**2
+    p *= 2
+
+  print("  %i <= %i**%i" % (n, bound, p))
+
+
 def test_eigenvalue(M, x):
   H = M - x*identity_matrix(QQ, M.ncols(), sparse=True)
   if H.is_singular():
     print("  Value %i is an eigenvalue of M with geometric multiplicity %i" %
       (x, H.ncols() - H.rank()))
 
+
 def matrix_to_rev_graph(M):
   assert M.ncols() == M.nrows()
   return DiGraph([range(M.ncols()), M.dict().keys()],
@@ -86,3 +123,9 @@ if __name__ == "__main__":
     if vec_[i, 0] > 0 and fin[0, i] > 0:
       print("    %s" % (S["variables"][i],))
 
+  print("Check that 16 is the largest eigenvalue")
+  print("  Calculating deflation of the matrix")
+  D = deflate_eigenvector(M, vec_)
+  print("  Bounding the spectral radius by 15")
+  bound_spectral_radius(D, 15)
+