#13990: Bug fix and small improvement of spanning_trees_count
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Reporter: azi | Owner: jason, ncohen, rlm
Type: defect | Status: needs_review
Priority: major | Milestone: sage-5.7
Component: graph theory | Resolution:
Keywords: | Work issues:
Report Upstream: N/A | Reviewers:
Authors: | Merged in:
Dependencies: | Stopgaps:
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Comment (by azi):
Damn. I haven't seen your comment..
A tree is a connected graph with n vertices and n-1 edges. There is no
subgraph of the empty graph having $-1$ edges and $0$ vertices hence there
is no spanning tree of the empty graph. I have also tested the function in
Mathematica which returns 0 as expected.
As for abs(). The matrix tree theorem states the following. Let L_{i,j} be
the Laplacian matrix of a graph G from which we remove the i'th column and
j'th row. Then the number of spanning trees is (-1)^{i+j}det(L_{i,j})
hence for i=j the expression simplifies to simply computing det(L_{i,j})
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Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/13990#comment:3>
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