Ticket #7184: trac_7184.patch

sage/graphs/graph.py

@@ -59 +59 @@
 
 -  Stephen Hartke (2009-07-26): Fixed bug in blocks_and_cut_vertices()
    that caused an incorrect result when the vertex 0 was a cut vertex.
+   
+-  Anders Jonsson (2009-10-10): Counting of spanning trees added.
 
 
 Graph Format
@@ -2102 +2104 @@
 
     num_edges = size
 
+    def spanning_trees(self, root_vertex=0):
+        """
+        Returns the number of spanning trees in a graph. In the case of a
+        digraph, use vertex number of the desired root vertex as argument.
+        Default is to set first vertex as root.
+        
+        This computation uses Kirchhoff's Matrix Tree Theorem [1] to calculate
+        the number of spanning trees. For complete graphs on n vertices the
+        result can also be reached using Cayley's formula: the number of
+        spanning trees are n^(n-2).
+        
+        For digraphs, the augmented Kirchhoff Matrix as defined in [2] is
+        used for calculations. Here the result is the number of out-trees
+        rooted at a specific vertex.
+        
+        INPUT:
+
+        - ``root_vertex`` - integer (default: 0, or the number of the first
+        vertex if vertex 0 not is in the graph) This is the vertex that will be
+        used as root for all spanning trees if the graph is a directed graph.
+       
+        REFERENCES:
+        
+        - [1] http://mathworld.wolfram.com/MatrixTreeTheorem.html
+
+        - [2] Lih-Hsing Hsu, Cheng-Kuan Lin, "Graph Theory and Interconnection
+        Networks"
+        
+        AUTHORS:
+              
+        - Anders Jonsson (2009-10-10)
+        
+        EXAMPLES::
+        
+            sage: G = graphs.PetersenGraph()
+            sage: G.spanning_trees()
+            2000
+            
+        ::
+        
+            sage: n = 11
+            sage: G = graphs.CompleteGraph(n)
+            sage: ST = G.spanning_trees()
+            sage: ST == n^(n-2)
+            True
+            
+        ::
+        
+            sage: M=matrix(3,3,[0,1,0,0,0,1,1,1,0])
+            sage: D=DiGraph(M)
+            sage: D.spanning_trees()
+            1
+            sage: D.spanning_trees(0)
+            1
+            sage: D.spanning_trees(2)
+            2
+        
+        """
+        if self.is_directed() == False:
+            M=self.kirchhoff_matrix()
+            M.subdivide(1,1)
+            M2 = M.subdivision(1,1)
+            return abs(M2.determinant())
+        else:
+            G=self.copy()
+            n = G.vertices()[G.order()-1] + 1
+            if root_vertex == 0:
+                root_vertex=G.vertices()[0]
+            if root_vertex not in G.vertices():
+                raise RuntimeError("Vertex (%s) not in the graph."%root_vertex)
+            G.add_vertex(n)
+            G.add_edge(n,root_vertex)
+            M=G.kirchhoff_matrix()
+            j=0
+            for i in G.vertices():
+                M[j,j]=G.in_degree(i)
+                j= j + 1
+            M.subdivide(G.order()-1,G.order()-1)
+            M2=M.subdivision(0,0)
+            return abs(M2.determinant())
+
     ### Planarity
 
     def is_planar(self, on_embedding=None, kuratowski=False, set_embedding=False, set_pos=False):