# scipy.sparse.csgraph.structural_rank¶

scipy.sparse.csgraph.structural_rank(graph)

Compute the structural rank of a graph (matrix) with a given sparsity pattern.

The structural rank of a matrix is the number of entries in the maximum transversal of the corresponding bipartite graph, and is an upper bound on the numerical rank of the matrix. A graph has full structural rank if it is possible to permute the elements to make the diagonal zero-free.

New in version 0.19.0.

Parameters
graphsparse matrix

Input sparse matrix.

Returns
rankint

The structural rank of the sparse graph.

References

1

I. S. Duff, “Computing the Structural Index”, SIAM J. Alg. Disc. Meth., Vol. 7, 594 (1986).

2

http://www.cise.ufl.edu/research/sparse/matrices/legend.html

Examples

>>> from scipy.sparse import csr_matrix
>>> from scipy.sparse.csgraph import structural_rank

>>> graph = [
... [0, 1, 2, 0],
... [1, 0, 0, 1],
... [2, 0, 0, 3],
... [0, 1, 3, 0]
... ]
>>> graph = csr_matrix(graph)
>>> print(graph)
(0, 1)    1
(0, 2)    2
(1, 0)    1
(1, 3)    1
(2, 0)    2
(2, 3)    3
(3, 1)    1
(3, 2)    3

>>> structural_rank(graph)
4


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