Implement counting of spanning trees for graphs and digraphs

[haaninjo]

This patch allows us to count the number of spanning trees in a simple graph, as well as the spanning out-trees from a user-defined root node in a digraph.

Method used: Kirchhoff's matrix tree theorem [1] and the Laplacian matrix for the simple graphs, and a variation of the same [2] in the directed case.

[1] http://en.wikipedia.org/wiki/Kirchhoff%27s_theorem

[2] corollary 4.4 in http://books.google.se/books?id=vbxdqhDKOSYC&printsec=frontcover&hl=en&source=gbs_navlinks_s

CC: @boothby

Component: graph theory

Author: Anders Jonsson, Nathann Cohen

Reviewer: Mike Hansen

Merged: sage-4.3.rc0

Issue created by migration from https://trac.sagemath.org/ticket/7184

[haaninjo]

count spanning trees of graphs

[haaninjo]

comment:1

Attachment: trac_7184.patch.gz

[haaninjo]

Description changed:

--- 
+++ 
@@ -3,5 +3,6 @@
 Method used: Kirchhoff's matrix tree theorem [1] and the Laplacian matrix for the simple graphs, and a variation of the same [2] in the directed case.
 
 
-[1] http://en.wikipedia.org/wiki/Kirchhoff%27s_theorem
+[1] http://en.wikipedia.org/wiki/Kirchhoff%27s_theorem 
+
 [2] corollary 4.4 in http://books.google.se/books?id=vbxdqhDKOSYC&printsec=frontcover&hl=en&source=gbs_navlinks_s

[nathanncohen]

comment:3

Here is a new patch based on yours, plus some modifications :

• I was not able at first to apply your patch, as it was based on an old version of Sage. This new version is now based on 4.1.2.rc0.

• I fixed some docstrings :
  * you set "math" mode with this sign : ( exemple :x^2` )
  * you set "sage" mode with this sign : ( exemple :graph.am()`` )

• I renamed the function from spanning_trees to spanning_tree_count as the first seemed to imply an iterator over spanning_trees

• Some details concerning the root_vertex too. Many graphs have vertices not among integers but among strings, or tuples, etc. Besides, it is not necessary to enumerate all the vertices to get the first one :

  ```
  v=G.vertices()[0]
  

  You can also do :
  
      ```
      v=G.vertex_iterator().next()
  

• You had to modify the adjacency matrix to fit the definition used in your references. This was perfectly fine, but also meant there was an error in the definition of kirchhoff_matrix for directed graph. This patch fixes this, plus some docstrings error in the kirchhoff_matrix function.

Short of these details, you algorithm works fine, thank you for having sent this patch !!

If you agree with this nex patch I am sending, you can set this ticket to positive_review

Thanks !

Nathann

Algorithm is OK, both for Graphs and Digraphs. You had to re-define

[nathanncohen]

comment:4

Ooops, sorry for the last line, which was not meant to be included in my message. Besides, I meant that you had to redefine the kirchhoff matrix, not the adjacency matrix.. Sorry :-)

[boothby]

comment:5

Ncohen, you've broken kirchhoff_matrix. According to the definition, row-sums of the Kirchhoff matrix should be zero. Here's the doctest before you changed it:

sage: G = DiGraph({1:{1:2,2:3}, 2:{1:4}}, weighted=True,sparse=True) 
sage: G.laplacian_matrix() 
[ 3 -3] 
[-4  4] 

here, the row-sums are zero.

But you actually had to change the doctest to make the code pass,

sage: G = DiGraph({1:{1:2,2:3}, 2:{1:4}}, weighted=True,sparse=True) 
sage: G.laplacian_matrix() 
[ 4 -3] 
[-4  3] 

but the row-sums aren't zero! Don't break math to make code work!