venv/lib/python3.12/site-packages/networkx/generators/harary_graph.py

"""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):
    r"""Return 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 $\lfloor 2m/n \rfloor$. [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 (default=nx.Graph)
        Graph type to create. 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.
    The implementation follows [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")

    # Get the floor of average node degree.
    d = 2 * m // n

    offset = d // 2
    H = nx.circulant_graph(n, range(1, offset + 1), create_using=create_using)

    half = n // 2
    if (n % 2 == 0) or (d % 2 == 0):
        # If d is odd; n must be even.
        if d % 2 == 1:
            # Add edges diagonally.
            H.add_edges_from((i, i + half) for i in range(half))

        r = 2 * m % n
        # Add remaining edges at offset + 1.
        H.add_edges_from((i, i + offset + 1) for i in range(r // 2))
    else:
        # Add the remaining m - n * offset edges between i and i + half.
        H.add_edges_from((i, (i + half) % n) for i in range(m - n * offset))

    return H


@nx._dispatchable(graphs=None, returns_graph=True)
def hkn_harary_graph(k, n, create_using=None):
    r"""Return 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 $\lceil kn/2 \rceil$ [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 (default=nx.Graph)
        Graph type to create. 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.
    The implementation follows [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:
        return nx.path_graph(n, create_using)

    offset = k // 2
    H = nx.circulant_graph(n, range(1, offset + 1), create_using=create_using)

    half = n // 2
    if (k % 2 == 0) or (n % 2 == 0):
        # If k is odd; n must be even.
        if k % 2 == 1:
            # Add edges diagonally.
            H.add_edges_from((i, i + half) for i in range(half))
    else:
        # Add half + 1 edges between i and i + half.
        H.add_edges_from((i, (i + half) % n) for i in range(half + 1))

    return H