Matching Polynomial Routine
This module contains the following methods:
matching_polynomial() | Computes the matching polynomial of a given graph |
complete_poly() | Compute the matching polynomial of the complete graph on ![]() |
AUTHORS:
REFERENCE:
[Godsil93] | (1, 2, 3) Chris Godsil (1993) Algebraic Combinatorics. |
Compute the matching polynomial of the complete graph on vertices.
INPUT:
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
Computes the matching polynomial of the graph .
If denotes the number of
-matchings (matchings with
edges)
in
, then the matching polynomial is defined as [Godsil93]:
INPUT:
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 is an edge of
,
is the result of deleting the edge
, and
is the result of deleting each vertex in
, then the matching
polynomial of
is equal to that of
minus that of
.
(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 :
Where is the complement of
, and
the complete graph
on
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