"""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.to_numpy_array(G, dtype=int) >>> M = sympy.SparseMatrix(A) >>> 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