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# Source code for networkx.generators.harary_graph

    """Generators for Harary graphs

    This module gives two generators for the Harary graph, which was
    introduced by the famous mathematician Frank Harary in his 1962 work [H]_.
    The first generator gives the Harary graph that maximizes the node
    connectivity with given number of nodes and given number of edges.
    The second generator gives the Harary graph that minimizes
    the number of edges in the graph with given node connectivity and
    number of nodes.

    References
    ----------
    .. [H] Harary, F. "The Maximum Connectivity of a Graph."
           Proc. Nat. Acad. Sci. USA 48, 1142-1146, 1962.

    """

    import networkx as nx
    from networkx.exception import NetworkXError

    __all__ = ["hnm_harary_graph", "hkn_harary_graph"]

    @nx._dispatchable(graphs=None, returns_graph=True)
    def hnm_harary_graph(n, m, create_using=None):
        """Returns the Harary graph with given numbers of nodes and edges.

        The Harary graph $H_{n,m}$ is the graph that maximizes node connectivity
        with $n$ nodes and $m$ edges.

        This maximum node connectivity is known to be floor($2m/n$). [1]_

        Parameters
        ----------
        n: integer
           The number of nodes the generated graph is to contain

        m: integer
           The number of edges the generated graph is to contain

        create_using : NetworkX graph constructor, optional Graph type
         to create (default=nx.Graph). If graph instance, then cleared
         before populated.

        Returns
        -------
        NetworkX graph
            The Harary graph $H_{n,m}$.

        See Also
        --------
        hkn_harary_graph

        Notes
        -----
        This algorithm runs in $O(m)$ time.
        It is implemented by following the Reference [2]_.

        References
        ----------
        .. [1] F. T. Boesch, A. Satyanarayana, and C. L. Suffel,
           "A Survey of Some Network Reliability Analysis and Synthesis Results,"
           Networks, pp. 99-107, 2009.

        .. [2] Harary, F. "The Maximum Connectivity of a Graph."
           Proc. Nat. Acad. Sci. USA 48, 1142-1146, 1962.
        """

        if n < 1:
            raise NetworkXError("The number of nodes must be >= 1!")
        if m < n - 1:
            raise NetworkXError("The number of edges must be >= n - 1 !")
        if m > n * (n - 1) // 2:
            raise NetworkXError("The number of edges must be <= n(n-1)/2")

        # Construct an empty graph with n nodes first
        H = nx.empty_graph(n, create_using)
        # Get the floor of average node degree
        d = 2 * m // n

        # Test the parity of n and d
        if (n % 2 == 0) or (d % 2 == 0):
            # Start with a regular graph of d degrees
            offset = d // 2
            for i in range(n):
                for j in range(1, offset + 1):
                    H.add_edge(i, (i - j) % n)
                    H.add_edge(i, (i + j) % n)
            if d & 1:
                # in case d is odd; n must be even in this case
                half = n // 2
                for i in range(half):
                    # add edges diagonally
                    H.add_edge(i, i + half)
            # Get the remainder of 2*m modulo n
            r = 2 * m % n
            if r > 0:
                # add remaining edges at offset+1
                for i in range(r // 2):
                    H.add_edge(i, i + offset + 1)
        else:
            # Start with a regular graph of (d - 1) degrees
            offset = (d - 1) // 2
            for i in range(n):
                for j in range(1, offset + 1):
                    H.add_edge(i, (i - j) % n)
                    H.add_edge(i, (i + j) % n)
            half = n // 2
            for i in range(m - n * offset):
                # add the remaining m - n*offset edges between i and i+half
                H.add_edge(i, (i + half) % n)

        return H

    @nx._dispatchable(graphs=None, returns_graph=True)
    def hkn_harary_graph(k, n, create_using=None):
        """Returns the Harary graph with given node connectivity and node number.

        The Harary graph $H_{k,n}$ is the graph that minimizes the number of
        edges needed with given node connectivity $k$ and node number $n$.

        This smallest number of edges is known to be ceil($kn/2$) [1]_.

        Parameters
        ----------
        k: integer
           The node connectivity of the generated graph

        n: integer
           The number of nodes the generated graph is to contain

        create_using : NetworkX graph constructor, optional Graph type
         to create (default=nx.Graph). If graph instance, then cleared
         before populated.

        Returns
        -------
        NetworkX graph
            The Harary graph $H_{k,n}$.

        See Also
        --------
        hnm_harary_graph

        Notes
        -----
        This algorithm runs in $O(kn)$ time.
        It is implemented by following the Reference [2]_.

        References
        ----------
        .. [1] Weisstein, Eric W. "Harary Graph." From MathWorld--A Wolfram Web
         Resource. http://mathworld.wolfram.com/HararyGraph.html.

        .. [2] Harary, F. "The Maximum Connectivity of a Graph."
          Proc. Nat. Acad. Sci. USA 48, 1142-1146, 1962.
        """

        if k < 1:
            raise NetworkXError("The node connectivity must be >= 1!")
        if n < k + 1:
            raise NetworkXError("The number of nodes must be >= k+1 !")

        # in case of connectivity 1, simply return the path graph
        if k == 1:
            H = nx.path_graph(n, create_using)
            return H

        # Construct an empty graph with n nodes first
        H = nx.empty_graph(n, create_using)

        # Test the parity of k and n
        if (k % 2 == 0) or (n % 2 == 0):
            # Construct a regular graph with k degrees
            offset = k // 2
            for i in range(n):
                for j in range(1, offset + 1):
                    H.add_edge(i, (i - j) % n)
                    H.add_edge(i, (i + j) % n)
            if k & 1:
                # odd degree; n must be even in this case
                half = n // 2
                for i in range(half):
                    # add edges diagonally
                    H.add_edge(i, i + half)
        else:
            # Construct a regular graph with (k - 1) degrees
            offset = (k - 1) // 2
            for i in range(n):
                for j in range(1, offset + 1):
                    H.add_edge(i, (i - j) % n)
                    H.add_edge(i, (i + j) % n)
            half = n // 2
            for i in range(half + 1):
                # add half+1 edges between i and i+half
                H.add_edge(i, (i + half) % n)

        return H