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This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.

Source code for networkx.generators.expanders

# -*- coding: utf-8 -*-
# Copyright 2014 "cheebee7i".
# Copyright 2014 "alexbrc".
# Copyright 2014 Jeffrey Finkelstein <jeffrey.finkelstein@gmail.com>.
"""Provides explicit constructions of expander graphs.

"""
import itertools
import networkx as nx

__all__ = ['margulis_gabber_galil_graph', 'chordal_cycle_graph']


# Other discrete torus expanders can be constructed by using the following edge
# sets. For more information, see Chapter 4, "Expander Graphs", in
# "Pseudorandomness", by Salil Vadhan.
#
# For a directed expander, add edges from (x, y) to:
#
#     (x, y),
#     ((x + 1) % n, y),
#     (x, (y + 1) % n),
#     (x, (x + y) % n),
#     (-y % n, x)
#
# For an undirected expander, add the reverse edges.
#
# Also appearing in the paper of Gabber and Galil:
#
#     (x, y),
#     (x, (x + y) % n),
#     (x, (x + y + 1) % n),
#     ((x + y) % n, y),
#     ((x + y + 1) % n, y)
#
# and:
#
#     (x, y),
#     ((x + 2*y) % n, y),
#     ((x + (2*y + 1)) % n, y),
#     ((x + (2*y + 2)) % n, y),
#     (x, (y + 2*x) % n),
#     (x, (y + (2*x + 1)) % n),
#     (x, (y + (2*x + 2)) % n),
#
[docs]def margulis_gabber_galil_graph(n, create_using=None): r"""Returns the Margulis-Gabber-Galil undirected MultiGraph on `n^2` nodes. The undirected MultiGraph is regular with degree `8`. Nodes are integer pairs. The second-largest eigenvalue of the adjacency matrix of the graph is at most `5 \sqrt{2}`, regardless of `n`. Parameters ---------- n : int Determines the number of nodes in the graph: `n^2`. create_using : NetworkX graph constructor, optional (default MultiGraph) Graph type to create. If graph instance, then cleared before populated. Returns ------- G : graph The constructed undirected multigraph. Raises ------ NetworkXError If the graph is directed or not a multigraph. """ G = nx.empty_graph(0, create_using, default=nx.MultiGraph) if G.is_directed() or not G.is_multigraph(): msg = "`create_using` must be an undirected multigraph." raise nx.NetworkXError(msg) for (x, y) in itertools.product(range(n), repeat=2): for (u, v) in (((x + 2 * y) % n, y), ((x + (2 * y + 1)) % n, y), (x, (y + 2 * x) % n), (x, (y + (2 * x + 1)) % n)): G.add_edge((x, y), (u, v)) G.graph['name'] = "margulis_gabber_galil_graph({0})".format(n) return G
[docs]def chordal_cycle_graph(p, create_using=None): """Returns the chordal cycle graph on `p` nodes. The returned graph is a cycle graph on `p` nodes with chords joining each vertex `x` to its inverse modulo `p`. This graph is a (mildly explicit) 3-regular expander [1]_. `p` *must* be a prime number. Parameters ---------- p : a prime number The number of vertices in the graph. This also indicates where the chordal edges in the cycle will be created. create_using : NetworkX graph constructor, optional (default=nx.Graph) Graph type to create. If graph instance, then cleared before populated. Returns ------- G : graph The constructed undirected multigraph. Raises ------ NetworkXError If `create_using` indicates directed or not a multigraph. References ---------- .. [1] Theorem 4.4.2 in A. Lubotzky. "Discrete groups, expanding graphs and invariant measures", volume 125 of Progress in Mathematics. Birkhäuser Verlag, Basel, 1994. """ G = nx.empty_graph(0, create_using, default=nx.MultiGraph) if G.is_directed() or not G.is_multigraph(): msg = "`create_using` must be an undirected multigraph." raise nx.NetworkXError(msg) for x in range(p): left = (x - 1) % p right = (x + 1) % p # Here we apply Fermat's Little Theorem to compute the multiplicative # inverse of x in Z/pZ. By Fermat's Little Theorem, # # x^p = x (mod p) # # Therefore, # # x * x^(p - 2) = 1 (mod p) # # The number 0 is a special case: we just let its inverse be itself. chord = pow(x, p - 2, p) if x > 0 else 0 for y in (left, right, chord): G.add_edge(x, y) G.graph['name'] = "chordal_cycle_graph({0})".format(p) return G