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# cython: binding=True """ Matching Polynomial
This module contains the following methods:
.. csv-table:: :class: contentstable :widths: 30, 70 :delim: |
:meth:`matching_polynomial` | Computes the matching polynomial of a given graph :meth:`complete_poly` | Compute the matching polynomial of the complete graph on `n` vertices.
AUTHORS:
- Robert Miller, Tom Boothby - original implementation
REFERENCE:
.. [Godsil93] Chris Godsil (1993) Algebraic Combinatorics.
Methods ------- """
#***************************************************************************** # Copyright (C) 2010 Robert Miller # # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 2 of the License, or # (at your option) any later version. # http://www.gnu.org/licenses/ #*****************************************************************************
from cysignals.memory cimport check_allocarray, sig_free from cysignals.signals cimport sig_on, sig_off
from sage.rings.polynomial.polynomial_ring import polygen from sage.rings.integer_ring import ZZ from sage.rings.integer cimport Integer from sage.misc.all import prod
from sage.libs.flint.fmpz cimport * from sage.libs.flint.fmpz_poly cimport *
x = polygen(ZZ, 'x')
def matching_polynomial(G, complement=True, name=None): """ Computes the matching polynomial of the graph `G`.
If `p(G, k)` denotes the number of `k`-matchings (matchings with `k` edges) in `G`, then the matching polynomial is defined as [Godsil93]_:
.. MATH::
\mu(x)=\sum_{k \geq 0} (-1)^k p(G,k) x^{n-2k}
INPUT:
- ``complement`` - (default: ``True``) whether to use Godsil's duality theorem to compute the matching polynomial from that of the graphs complement (see ALGORITHM).
- ``name`` - optional string for the variable name in the polynomial
.. NOTE::
The ``complement`` option uses matching polynomials of complete graphs, which are cached. So if you are crazy enough to try computing the matching polynomial on a graph with millions of vertices, you might not want to use this option, since it will end up caching millions of polynomials of degree in the millions.
ALGORITHM:
The algorithm used is a recursive one, based on the following observation [Godsil93]_:
- If `e` is an edge of `G`, `G'` is the result of deleting the edge `e`, and `G''` is the result of deleting each vertex in `e`, then the matching polynomial of `G` is equal to that of `G'` minus that of `G''`.
(the algorithm actually computes the *signless* matching polynomial, for which the recursion is the same when one replaces the substraction by an addition. It is then converted into the matching polynomial and returned)
Depending on the value of ``complement``, Godsil's duality theorem [Godsil93]_ can also be used to compute `\mu(x)` :
.. MATH::
\mu(\overline{G}, x) = \sum_{k \geq 0} p(G,k) \mu( K_{n-2k}, x)
Where `\overline{G}` is the complement of `G`, and `K_n` the complete graph on `n` vertices.
EXAMPLES::
sage: g = graphs.PetersenGraph() sage: g.matching_polynomial() x^10 - 15*x^8 + 75*x^6 - 145*x^4 + 90*x^2 - 6 sage: g.matching_polynomial(complement=False) x^10 - 15*x^8 + 75*x^6 - 145*x^4 + 90*x^2 - 6 sage: g.matching_polynomial(name='tom') tom^10 - 15*tom^8 + 75*tom^6 - 145*tom^4 + 90*tom^2 - 6 sage: g = Graph() sage: L = [graphs.RandomGNP(8, .3) for i in range(1, 6)] sage: prod([h.matching_polynomial() for h in L]) == sum(L, g).matching_polynomial() # long time (up to 10s on sage.math, 2011) True
::
sage: for i in range(1, 12): # long time (10s on sage.math, 2011) ....: for t in graphs.trees(i): ....: if t.matching_polynomial() != t.characteristic_polynomial(): ....: raise RuntimeError('bug for a tree A of size {0}'.format(i)) ....: c = t.complement() ....: if c.matching_polynomial(complement=False) != c.matching_polynomial(): ....: raise RuntimeError('bug for a tree B of size {0}'.format(i))
::
sage: from sage.graphs.matchpoly import matching_polynomial sage: matching_polynomial(graphs.CompleteGraph(0)) 1 sage: matching_polynomial(graphs.CompleteGraph(1)) x sage: matching_polynomial(graphs.CompleteGraph(2)) x^2 - 1 sage: matching_polynomial(graphs.CompleteGraph(3)) x^3 - 3*x sage: matching_polynomial(graphs.CompleteGraph(4)) x^4 - 6*x^2 + 3 sage: matching_polynomial(graphs.CompleteGraph(5)) x^5 - 10*x^3 + 15*x sage: matching_polynomial(graphs.CompleteGraph(6)) x^6 - 15*x^4 + 45*x^2 - 15 sage: matching_polynomial(graphs.CompleteGraph(7)) x^7 - 21*x^5 + 105*x^3 - 105*x sage: matching_polynomial(graphs.CompleteGraph(8)) x^8 - 28*x^6 + 210*x^4 - 420*x^2 + 105 sage: matching_polynomial(graphs.CompleteGraph(9)) x^9 - 36*x^7 + 378*x^5 - 1260*x^3 + 945*x sage: matching_polynomial(graphs.CompleteGraph(10)) x^10 - 45*x^8 + 630*x^6 - 3150*x^4 + 4725*x^2 - 945 sage: matching_polynomial(graphs.CompleteGraph(11)) x^11 - 55*x^9 + 990*x^7 - 6930*x^5 + 17325*x^3 - 10395*x sage: matching_polynomial(graphs.CompleteGraph(12)) x^12 - 66*x^10 + 1485*x^8 - 13860*x^6 + 51975*x^4 - 62370*x^2 + 10395 sage: matching_polynomial(graphs.CompleteGraph(13)) x^13 - 78*x^11 + 2145*x^9 - 25740*x^7 + 135135*x^5 - 270270*x^3 + 135135*x
::
sage: G = Graph({0:[1,2], 1:[2]}) sage: matching_polynomial(G) x^3 - 3*x sage: G = Graph({0:[1,2]}) sage: matching_polynomial(G) x^3 - 2*x sage: G = Graph({0:[1], 2:[]}) sage: matching_polynomial(G) x^3 - x sage: G = Graph({0:[], 1:[], 2:[]}) sage: matching_polynomial(G) x^3
::
sage: matching_polynomial(graphs.CompleteGraph(0), complement=False) 1 sage: matching_polynomial(graphs.CompleteGraph(1), complement=False) x sage: matching_polynomial(graphs.CompleteGraph(2), complement=False) x^2 - 1 sage: matching_polynomial(graphs.CompleteGraph(3), complement=False) x^3 - 3*x sage: matching_polynomial(graphs.CompleteGraph(4), complement=False) x^4 - 6*x^2 + 3 sage: matching_polynomial(graphs.CompleteGraph(5), complement=False) x^5 - 10*x^3 + 15*x sage: matching_polynomial(graphs.CompleteGraph(6), complement=False) x^6 - 15*x^4 + 45*x^2 - 15 sage: matching_polynomial(graphs.CompleteGraph(7), complement=False) x^7 - 21*x^5 + 105*x^3 - 105*x sage: matching_polynomial(graphs.CompleteGraph(8), complement=False) x^8 - 28*x^6 + 210*x^4 - 420*x^2 + 105 sage: matching_polynomial(graphs.CompleteGraph(9), complement=False) x^9 - 36*x^7 + 378*x^5 - 1260*x^3 + 945*x sage: matching_polynomial(graphs.CompleteGraph(10), complement=False) x^10 - 45*x^8 + 630*x^6 - 3150*x^4 + 4725*x^2 - 945 sage: matching_polynomial(graphs.CompleteGraph(11), complement=False) x^11 - 55*x^9 + 990*x^7 - 6930*x^5 + 17325*x^3 - 10395*x sage: matching_polynomial(graphs.CompleteGraph(12), complement=False) x^12 - 66*x^10 + 1485*x^8 - 13860*x^6 + 51975*x^4 - 62370*x^2 + 10395 sage: matching_polynomial(graphs.CompleteGraph(13), complement=False) x^13 - 78*x^11 + 2145*x^9 - 25740*x^7 + 135135*x^5 - 270270*x^3 + 135135*x
TESTS:
Non-integer labels should work, (:trac:`15545`)::
sage: G = Graph(10); sage: G.add_vertex((0,1)) sage: G.add_vertex('X') sage: G.matching_polynomial() x^12 """
cdef int nverts, nedges, i, j, cur cdef int *edges1 cdef int *edges2 cdef int *edges_mem cdef int **edges cdef fmpz_poly_t pol
raise NotImplementedError
# Using Godsil's duality theorem when the graph is dense
# Relabelling the vertices of the graph as [0...n-1] so that they are sorted # in increasing order of degree
# Initialization of pol, edges* variables.
# The edges_mem table is of size (2 * nedges * nedges), and is to be read as # nedges blocks of size (2 * nedges). These blocks of size (2 * nedges) are # themselves to be read as two blocks of length nedges, addressed as edges1 # and edges2. # # Only the first block of size (2 * nedges) is here filled. The function # delete_and_add will need the rest of the memory.
# Computing the signless matching polynomial
# Building the actual matching polynomial
cdef Integer c_ZZ
# The following is a cache of complete graph matching polynomials.
cdef list complete_matching_polys = [x.parent().one(), x]
def complete_poly(n): """ Compute the matching polynomial of the complete graph on `n` vertices.
INPUT:
- ``n`` -- order of the complete graph
.. TODO::
This code could probably be made more efficient by using FLINT polynomials and being written in Cython, using an array of fmpz_poly_t pointers or something... Right now just about the whole complement optimization is written in Python, and could be easily sped up.
EXAMPLES::
sage: from sage.graphs.matchpoly import complete_poly sage: f = complete_poly(10) sage: f x^10 - 45*x^8 + 630*x^6 - 3150*x^4 + 4725*x^2 - 945 sage: f = complete_poly(20) sage: f[8] 1309458150 sage: f = complete_poly(1000) sage: len(str(f)) 406824
TESTS:
Checking the numerical results up to 20::
sage: from sage.functions.orthogonal_polys import hermite sage: p = lambda n: 2^(-n/2)*hermite(n, x/sqrt(2)) sage: all(p(i) == complete_poly(i) for i in range(2, 20)) True """ # global complete_matching_polys # if we do eventually make it a C array...
cdef void delete_and_add(int **edges, int nverts, int nedges, int totverts, int depth, fmpz_poly_t pol): """ Add matching polynomial to pol via recursion.
NOTE : at the end of this function, pol represents the *SIGNLESS* matching polynomial. """ cdef int i, j, k, edge1, edge2, new_edge1, new_edge2, new_nedges cdef int *edges1 cdef int *edges2 cdef int *new_edges1 cdef int *new_edges2 cdef fmpz * coeff
else: fmpz_poly_set_coeff_ui(pol, 1, nedges) else:
else:
# The last edge is (edge1, edge2)
# The new edges are all the edges that are not incident with (edge1, edge2)
# (the edges # are sorted by increasing order of their first component)
# since edge2 > edge1 > edges1[i], only need to check edges2[i]
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