306 lines
11 KiB
Python
306 lines
11 KiB
Python
"""Provides algorithms supporting the computation of graph polynomials.
|
|
|
|
Graph polynomials are polynomial-valued graph invariants that encode a wide
|
|
variety of structural information. Examples include the Tutte polynomial,
|
|
chromatic polynomial, characteristic polynomial, and matching polynomial. An
|
|
extensive treatment is provided in [1]_.
|
|
|
|
For a simple example, the `~sympy.matrices.matrices.MatrixDeterminant.charpoly`
|
|
method can be used to compute the characteristic polynomial from the adjacency
|
|
matrix of a graph. Consider the complete graph ``K_4``:
|
|
|
|
>>> import sympy
|
|
>>> x = sympy.Symbol("x")
|
|
>>> G = nx.complete_graph(4)
|
|
>>> A = nx.adjacency_matrix(G)
|
|
>>> M = sympy.SparseMatrix(A.todense())
|
|
>>> M.charpoly(x).as_expr()
|
|
x**4 - 6*x**2 - 8*x - 3
|
|
|
|
|
|
.. [1] Y. Shi, M. Dehmer, X. Li, I. Gutman,
|
|
"Graph Polynomials"
|
|
"""
|
|
from collections import deque
|
|
|
|
import networkx as nx
|
|
from networkx.utils import not_implemented_for
|
|
|
|
__all__ = ["tutte_polynomial", "chromatic_polynomial"]
|
|
|
|
|
|
@not_implemented_for("directed")
|
|
@nx._dispatchable
|
|
def tutte_polynomial(G):
|
|
r"""Returns the Tutte polynomial of `G`
|
|
|
|
This function computes the Tutte polynomial via an iterative version of
|
|
the deletion-contraction algorithm.
|
|
|
|
The Tutte polynomial `T_G(x, y)` is a fundamental graph polynomial invariant in
|
|
two variables. It encodes a wide array of information related to the
|
|
edge-connectivity of a graph; "Many problems about graphs can be reduced to
|
|
problems of finding and evaluating the Tutte polynomial at certain values" [1]_.
|
|
In fact, every deletion-contraction-expressible feature of a graph is a
|
|
specialization of the Tutte polynomial [2]_ (see Notes for examples).
|
|
|
|
There are several equivalent definitions; here are three:
|
|
|
|
Def 1 (rank-nullity expansion): For `G` an undirected graph, `n(G)` the
|
|
number of vertices of `G`, `E` the edge set of `G`, `V` the vertex set of
|
|
`G`, and `c(A)` the number of connected components of the graph with vertex
|
|
set `V` and edge set `A` [3]_:
|
|
|
|
.. math::
|
|
|
|
T_G(x, y) = \sum_{A \in E} (x-1)^{c(A) - c(E)} (y-1)^{c(A) + |A| - n(G)}
|
|
|
|
Def 2 (spanning tree expansion): Let `G` be an undirected graph, `T` a spanning
|
|
tree of `G`, and `E` the edge set of `G`. Let `E` have an arbitrary strict
|
|
linear order `L`. Let `B_e` be the unique minimal nonempty edge cut of
|
|
$E \setminus T \cup {e}$. An edge `e` is internally active with respect to
|
|
`T` and `L` if `e` is the least edge in `B_e` according to the linear order
|
|
`L`. The internal activity of `T` (denoted `i(T)`) is the number of edges
|
|
in $E \setminus T$ that are internally active with respect to `T` and `L`.
|
|
Let `P_e` be the unique path in $T \cup {e}$ whose source and target vertex
|
|
are the same. An edge `e` is externally active with respect to `T` and `L`
|
|
if `e` is the least edge in `P_e` according to the linear order `L`. The
|
|
external activity of `T` (denoted `e(T)`) is the number of edges in
|
|
$E \setminus T$ that are externally active with respect to `T` and `L`.
|
|
Then [4]_ [5]_:
|
|
|
|
.. math::
|
|
|
|
T_G(x, y) = \sum_{T \text{ a spanning tree of } G} x^{i(T)} y^{e(T)}
|
|
|
|
Def 3 (deletion-contraction recurrence): For `G` an undirected graph, `G-e`
|
|
the graph obtained from `G` by deleting edge `e`, `G/e` the graph obtained
|
|
from `G` by contracting edge `e`, `k(G)` the number of cut-edges of `G`,
|
|
and `l(G)` the number of self-loops of `G`:
|
|
|
|
.. math::
|
|
T_G(x, y) = \begin{cases}
|
|
x^{k(G)} y^{l(G)}, & \text{if all edges are cut-edges or self-loops} \\
|
|
T_{G-e}(x, y) + T_{G/e}(x, y), & \text{otherwise, for an arbitrary edge $e$ not a cut-edge or loop}
|
|
\end{cases}
|
|
|
|
Parameters
|
|
----------
|
|
G : NetworkX graph
|
|
|
|
Returns
|
|
-------
|
|
instance of `sympy.core.add.Add`
|
|
A Sympy expression representing the Tutte polynomial for `G`.
|
|
|
|
Examples
|
|
--------
|
|
>>> C = nx.cycle_graph(5)
|
|
>>> nx.tutte_polynomial(C)
|
|
x**4 + x**3 + x**2 + x + y
|
|
|
|
>>> D = nx.diamond_graph()
|
|
>>> nx.tutte_polynomial(D)
|
|
x**3 + 2*x**2 + 2*x*y + x + y**2 + y
|
|
|
|
Notes
|
|
-----
|
|
Some specializations of the Tutte polynomial:
|
|
|
|
- `T_G(1, 1)` counts the number of spanning trees of `G`
|
|
- `T_G(1, 2)` counts the number of connected spanning subgraphs of `G`
|
|
- `T_G(2, 1)` counts the number of spanning forests in `G`
|
|
- `T_G(0, 2)` counts the number of strong orientations of `G`
|
|
- `T_G(2, 0)` counts the number of acyclic orientations of `G`
|
|
|
|
Edge contraction is defined and deletion-contraction is introduced in [6]_.
|
|
Combinatorial meaning of the coefficients is introduced in [7]_.
|
|
Universality, properties, and applications are discussed in [8]_.
|
|
|
|
Practically, up-front computation of the Tutte polynomial may be useful when
|
|
users wish to repeatedly calculate edge-connectivity-related information
|
|
about one or more graphs.
|
|
|
|
References
|
|
----------
|
|
.. [1] M. Brandt,
|
|
"The Tutte Polynomial."
|
|
Talking About Combinatorial Objects Seminar, 2015
|
|
https://math.berkeley.edu/~brandtm/talks/tutte.pdf
|
|
.. [2] A. Björklund, T. Husfeldt, P. Kaski, M. Koivisto,
|
|
"Computing the Tutte polynomial in vertex-exponential time"
|
|
49th Annual IEEE Symposium on Foundations of Computer Science, 2008
|
|
https://ieeexplore.ieee.org/abstract/document/4691000
|
|
.. [3] Y. Shi, M. Dehmer, X. Li, I. Gutman,
|
|
"Graph Polynomials," p. 14
|
|
.. [4] Y. Shi, M. Dehmer, X. Li, I. Gutman,
|
|
"Graph Polynomials," p. 46
|
|
.. [5] A. Nešetril, J. Goodall,
|
|
"Graph invariants, homomorphisms, and the Tutte polynomial"
|
|
https://iuuk.mff.cuni.cz/~andrew/Tutte.pdf
|
|
.. [6] D. B. West,
|
|
"Introduction to Graph Theory," p. 84
|
|
.. [7] G. Coutinho,
|
|
"A brief introduction to the Tutte polynomial"
|
|
Structural Analysis of Complex Networks, 2011
|
|
https://homepages.dcc.ufmg.br/~gabriel/seminars/coutinho_tuttepolynomial_seminar.pdf
|
|
.. [8] J. A. Ellis-Monaghan, C. Merino,
|
|
"Graph polynomials and their applications I: The Tutte polynomial"
|
|
Structural Analysis of Complex Networks, 2011
|
|
https://arxiv.org/pdf/0803.3079.pdf
|
|
"""
|
|
import sympy
|
|
|
|
x = sympy.Symbol("x")
|
|
y = sympy.Symbol("y")
|
|
stack = deque()
|
|
stack.append(nx.MultiGraph(G))
|
|
|
|
polynomial = 0
|
|
while stack:
|
|
G = stack.pop()
|
|
bridges = set(nx.bridges(G))
|
|
|
|
e = None
|
|
for i in G.edges:
|
|
if (i[0], i[1]) not in bridges and i[0] != i[1]:
|
|
e = i
|
|
break
|
|
if not e:
|
|
loops = list(nx.selfloop_edges(G, keys=True))
|
|
polynomial += x ** len(bridges) * y ** len(loops)
|
|
else:
|
|
# deletion-contraction
|
|
C = nx.contracted_edge(G, e, self_loops=True)
|
|
C.remove_edge(e[0], e[0])
|
|
G.remove_edge(*e)
|
|
stack.append(G)
|
|
stack.append(C)
|
|
return sympy.simplify(polynomial)
|
|
|
|
|
|
@not_implemented_for("directed")
|
|
@nx._dispatchable
|
|
def chromatic_polynomial(G):
|
|
r"""Returns the chromatic polynomial of `G`
|
|
|
|
This function computes the chromatic polynomial via an iterative version of
|
|
the deletion-contraction algorithm.
|
|
|
|
The chromatic polynomial `X_G(x)` is a fundamental graph polynomial
|
|
invariant in one variable. Evaluating `X_G(k)` for an natural number `k`
|
|
enumerates the proper k-colorings of `G`.
|
|
|
|
There are several equivalent definitions; here are three:
|
|
|
|
Def 1 (explicit formula):
|
|
For `G` an undirected graph, `c(G)` the number of connected components of
|
|
`G`, `E` the edge set of `G`, and `G(S)` the spanning subgraph of `G` with
|
|
edge set `S` [1]_:
|
|
|
|
.. math::
|
|
|
|
X_G(x) = \sum_{S \subseteq E} (-1)^{|S|} x^{c(G(S))}
|
|
|
|
|
|
Def 2 (interpolating polynomial):
|
|
For `G` an undirected graph, `n(G)` the number of vertices of `G`, `k_0 = 0`,
|
|
and `k_i` the number of distinct ways to color the vertices of `G` with `i`
|
|
unique colors (for `i` a natural number at most `n(G)`), `X_G(x)` is the
|
|
unique Lagrange interpolating polynomial of degree `n(G)` through the points
|
|
`(0, k_0), (1, k_1), \dots, (n(G), k_{n(G)})` [2]_.
|
|
|
|
|
|
Def 3 (chromatic recurrence):
|
|
For `G` an undirected graph, `G-e` the graph obtained from `G` by deleting
|
|
edge `e`, `G/e` the graph obtained from `G` by contracting edge `e`, `n(G)`
|
|
the number of vertices of `G`, and `e(G)` the number of edges of `G` [3]_:
|
|
|
|
.. math::
|
|
X_G(x) = \begin{cases}
|
|
x^{n(G)}, & \text{if $e(G)=0$} \\
|
|
X_{G-e}(x) - X_{G/e}(x), & \text{otherwise, for an arbitrary edge $e$}
|
|
\end{cases}
|
|
|
|
This formulation is also known as the Fundamental Reduction Theorem [4]_.
|
|
|
|
|
|
Parameters
|
|
----------
|
|
G : NetworkX graph
|
|
|
|
Returns
|
|
-------
|
|
instance of `sympy.core.add.Add`
|
|
A Sympy expression representing the chromatic polynomial for `G`.
|
|
|
|
Examples
|
|
--------
|
|
>>> C = nx.cycle_graph(5)
|
|
>>> nx.chromatic_polynomial(C)
|
|
x**5 - 5*x**4 + 10*x**3 - 10*x**2 + 4*x
|
|
|
|
>>> G = nx.complete_graph(4)
|
|
>>> nx.chromatic_polynomial(G)
|
|
x**4 - 6*x**3 + 11*x**2 - 6*x
|
|
|
|
Notes
|
|
-----
|
|
Interpretation of the coefficients is discussed in [5]_. Several special
|
|
cases are listed in [2]_.
|
|
|
|
The chromatic polynomial is a specialization of the Tutte polynomial; in
|
|
particular, ``X_G(x) = T_G(x, 0)`` [6]_.
|
|
|
|
The chromatic polynomial may take negative arguments, though evaluations
|
|
may not have chromatic interpretations. For instance, ``X_G(-1)`` enumerates
|
|
the acyclic orientations of `G` [7]_.
|
|
|
|
References
|
|
----------
|
|
.. [1] D. B. West,
|
|
"Introduction to Graph Theory," p. 222
|
|
.. [2] E. W. Weisstein
|
|
"Chromatic Polynomial"
|
|
MathWorld--A Wolfram Web Resource
|
|
https://mathworld.wolfram.com/ChromaticPolynomial.html
|
|
.. [3] D. B. West,
|
|
"Introduction to Graph Theory," p. 221
|
|
.. [4] J. Zhang, J. Goodall,
|
|
"An Introduction to Chromatic Polynomials"
|
|
https://math.mit.edu/~apost/courses/18.204_2018/Julie_Zhang_paper.pdf
|
|
.. [5] R. C. Read,
|
|
"An Introduction to Chromatic Polynomials"
|
|
Journal of Combinatorial Theory, 1968
|
|
https://math.berkeley.edu/~mrklug/ReadChromatic.pdf
|
|
.. [6] W. T. Tutte,
|
|
"Graph-polynomials"
|
|
Advances in Applied Mathematics, 2004
|
|
https://www.sciencedirect.com/science/article/pii/S0196885803000411
|
|
.. [7] R. P. Stanley,
|
|
"Acyclic orientations of graphs"
|
|
Discrete Mathematics, 2006
|
|
https://math.mit.edu/~rstan/pubs/pubfiles/18.pdf
|
|
"""
|
|
import sympy
|
|
|
|
x = sympy.Symbol("x")
|
|
stack = deque()
|
|
stack.append(nx.MultiGraph(G, contraction_idx=0))
|
|
|
|
polynomial = 0
|
|
while stack:
|
|
G = stack.pop()
|
|
edges = list(G.edges)
|
|
if not edges:
|
|
polynomial += (-1) ** G.graph["contraction_idx"] * x ** len(G)
|
|
else:
|
|
e = edges[0]
|
|
C = nx.contracted_edge(G, e, self_loops=True)
|
|
C.graph["contraction_idx"] = G.graph["contraction_idx"] + 1
|
|
C.remove_edge(e[0], e[0])
|
|
G.remove_edge(*e)
|
|
stack.append(G)
|
|
stack.append(C)
|
|
return polynomial
|