def _is_valid_digraph_edge_set( edges, frozen=0 ): """ Returns True if the input data is the edge set of a digraph for a quiver (no loops, no 2-cycles, edge-labels of the specified format), and returns False otherwise. INPUT: - ``frozen`` -- (integer; default:0) The number of frozen vertices. EXAMPLES:: sage: from sage.combinat.cluster_algebra_quiver.mutation_class import _is_valid_digraph_edge_set sage: _is_valid_digraph_edge_set( [[0,1,'a'],[2,3,(1,-1)]] ) The given digraph has edge labels which are not integral or integral 2-tuples. False sage: _is_valid_digraph_edge_set( [[0,1],[2,3,(1,-1)]] ) True sage: _is_valid_digraph_edge_set( [[0,1,'a'],[2,3,(1,-1)],[3,2,(1,-1)]] ) The given digraph or edge list contains oriented 2-cycles. False """ try: dg = DiGraph() dg.allow_multiple_edges(True) dg.add_edges( edges ) # checks if the digraph contains loops if dg.has_loops(): print "The given digraph or edge list contains loops." return False # checks if the digraph contains oriented 2-cycles if _has_two_cycles( dg ): print "The given digraph or edge list contains oriented 2-cycles." return False # checks if all edge labels are 'None', positive integers or tuples of positive integers if not all( i == None or ( i in ZZ and i > 0 ) or ( type(i) == tuple and len(i) == 2 and i[0] in ZZ and i[1] in ZZ ) for i in dg.edge_labels() ): print "The given digraph has edge labels which are not integral or integral 2-tuples." return False # checks if all edge labels for multiple edges are 'None' or positive integers if dg.has_multiple_edges(): for e in set( dg.multiple_edges(labels=False) ): if not all( i == None or ( i in ZZ and i > 0 ) for i in dg.edge_label( e[0], e[1] ) ): print "The given digraph or edge list contains multiple edges with non-integral labels." return False n = dg.order() - frozen if n < 0: print "The number of frozen variables is larger than the number of vertices." return False if [ e for e in dg.edges(labels=False) if e[0] >= n] <> []: print "The given digraph or edge list contains edges within the frozen vertices." return False return True except StandardError: print "Could not even build a digraph from the input data." return False
def _digraph_mutate( dg, k, n, m ): """ Returns a digraph obtained from dg with n+m vertices by mutating at vertex k. INPUT: - ``dg`` -- a digraph with integral edge labels with ``n+m`` vertices - ``k`` -- the vertex at which ``dg`` is mutated EXAMPLES:: sage: from sage.combinat.cluster_algebra_quiver.mutation_class import _digraph_mutate sage: dg = ClusterQuiver(['A',4]).digraph() sage: dg.edges() [(0, 1, (1, -1)), (2, 1, (1, -1)), (2, 3, (1, -1))] sage: _digraph_mutate(dg,2,4,0).edges() [(0, 1, (1, -1)), (1, 2, (1, -1)), (3, 2, (1, -1))] """ edges = dict( ((v1,v2),label) for v1,v2,label in dg._backend.iterator_in_edges(dg,True) ) in_edges = [ (v1,v2,edges[(v1,v2)]) for (v1,v2) in edges if v2 == k ] out_edges = [ (v1,v2,edges[(v1,v2)]) for (v1,v2) in edges if v1 == k ] in_edges_new = [ (v2,v1,(-label[1],-label[0])) for (v1,v2,label) in in_edges ] out_edges_new = [ (v2,v1,(-label[1],-label[0])) for (v1,v2,label) in out_edges ] diag_edges_new = [] diag_edges_del = [] for (v1,v2,label1) in in_edges: for (w1,w2,label2) in out_edges: l11,l12 = label1 l21,l22 = label2 if (v1,w2) in edges: diag_edges_del.append( (v1,w2,edges[(v1,w2)]) ) a,b = edges[(v1,w2)] a,b = a+l11*l21, b-l12*l22 diag_edges_new.append( (v1,w2,(a,b)) ) elif (w2,v1) in edges: diag_edges_del.append( (w2,v1,edges[(w2,v1)]) ) a,b = edges[(w2,v1)] a,b = b+l11*l21, a-l12*l22 if a<0: diag_edges_new.append( (w2,v1,(b,a)) ) elif a>0: diag_edges_new.append( (v1,w2,(a,b)) ) else: a,b = l11*l21,-l12*l22 diag_edges_new.append( (v1,w2,(a,b)) ) del_edges = in_edges + out_edges + diag_edges_del new_edges = in_edges_new + out_edges_new + diag_edges_new new_edges += [ (v1,v2,edges[(v1,v2)]) for (v1,v2) in edges if not (v1,v2,edges[(v1,v2)]) in del_edges ] dg_new = DiGraph() for v1,v2,label in new_edges: dg_new._backend.add_edge(v1,v2,label,True) if dg_new.order() < n+m: dg_new_vertices = [ v for v in dg_new ] for i in [ v for v in dg if v not in dg_new_vertices ]: dg_new.add_vertex(i) return dg_new
def _matrix_to_digraph( M ): """ Returns the digraph obtained from the matrix ``M``. In order to generate a quiver, we assume that ``M`` is skew-symmetrizable. EXAMPLES:: sage: from sage.combinat.cluster_algebra_quiver.mutation_class import _matrix_to_digraph sage: _matrix_to_digraph(matrix(3,[0,1,0,-1,0,-1,0,1,0])) Digraph on 3 vertices """ n = M.ncols() dg = DiGraph(sparse=True) for i,j in M.nonzero_positions(): if i >= n: a,b = M[i,j],-M[i,j] else: a,b = M[i,j],M[j,i] if a > 0: dg._backend.add_edge(i,j,(a,b),True) elif i >= n: dg._backend.add_edge(j,i,(-a,-b),True) if dg.order() < M.nrows(): for i in [ index for index in xrange(M.nrows()) if index not in dg ]: dg.add_vertex(i) return dg