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# Source code for networkx.algorithms.polynomials

    """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