"""
=======================
Distance-regular graphs
=======================
"""
from collections import defaultdict
from itertools import combinations_with_replacement
from math import log
import networkx as nx
from networkx.utils import not_implemented_for
from .distance_measures import diameter
__all__ = [
"is_distance_regular",
"is_strongly_regular",
"intersection_array",
"global_parameters",
]
[docs]
@nx._dispatchable
def is_distance_regular(G):
"""Returns True if the graph is distance regular, False otherwise.
A connected graph G is distance-regular if for any nodes x,y
and any integers i,j=0,1,...,d (where d is the graph
diameter), the number of vertices at distance i from x and
distance j from y depends only on i,j and the graph distance
between x and y, independently of the choice of x and y.
Parameters
----------
G: Networkx graph (undirected)
Returns
-------
bool
True if the graph is Distance Regular, False otherwise
Examples
--------
>>> G = nx.hypercube_graph(6)
>>> nx.is_distance_regular(G)
True
See Also
--------
intersection_array, global_parameters
Notes
-----
For undirected and simple graphs only
References
----------
.. [1] Brouwer, A. E.; Cohen, A. M.; and Neumaier, A.
Distance-Regular Graphs. New York: Springer-Verlag, 1989.
.. [2] Weisstein, Eric W. "Distance-Regular Graph."
http://mathworld.wolfram.com/Distance-RegularGraph.html
"""
try:
intersection_array(G)
return True
except nx.NetworkXError:
return False
[docs]
def global_parameters(b, c):
"""Returns global parameters for a given intersection array.
Given a distance-regular graph G with diameter d and integers b_i,
c_i,i = 0,....,d such that for any 2 vertices x,y in G at a distance
i=d(x,y), there are exactly c_i neighbors of y at a distance of i-1 from x
and b_i neighbors of y at a distance of i+1 from x.
Thus, a distance regular graph has the global parameters,
[[c_0,a_0,b_0],[c_1,a_1,b_1],......,[c_d,a_d,b_d]] for the
intersection array [b_0,b_1,.....b_{d-1};c_1,c_2,.....c_d]
where a_i+b_i+c_i=k , k= degree of every vertex.
Parameters
----------
b : list
c : list
Returns
-------
iterable
An iterable over three tuples.
Examples
--------
>>> G = nx.dodecahedral_graph()
>>> b, c = nx.intersection_array(G)
>>> list(nx.global_parameters(b, c))
[(0, 0, 3), (1, 0, 2), (1, 1, 1), (1, 1, 1), (2, 0, 1), (3, 0, 0)]
References
----------
.. [1] Weisstein, Eric W. "Global Parameters."
From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/GlobalParameters.html
See Also
--------
intersection_array
"""
return ((y, b[0] - x - y, x) for x, y in zip(b + [0], [0] + c))
[docs]
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatchable
def intersection_array(G):
"""Returns the intersection array of a distance-regular graph.
Given a distance-regular graph G with integers b_i, c_i,i = 0,....,d
such that for any 2 vertices x,y in G at a distance i=d(x,y), there
are exactly c_i neighbors of y at a distance of i-1 from x and b_i
neighbors of y at a distance of i+1 from x.
A distance regular graph's intersection array is given by,
[b_0,b_1,.....b_{d-1};c_1,c_2,.....c_d]
Parameters
----------
G: Networkx graph (undirected)
Returns
-------
b,c: tuple of lists
Examples
--------
>>> G = nx.icosahedral_graph()
>>> nx.intersection_array(G)
([5, 2, 1], [1, 2, 5])
References
----------
.. [1] Weisstein, Eric W. "Intersection Array."
From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/IntersectionArray.html
See Also
--------
global_parameters
"""
# the input graph is very unlikely to be distance-regular: here are the
# number a(n) of connected simple graphs, and the number b(n) of
# distance-regular graphs among them:
#
# n | 1 2 3 4 5 6 7 8 9 10
# -----+------------------------------------------------------------------
# a(n) | 1 1 2 6 21 112 853 11117 261080 11716571 https://oeis.org/A001349
# b(n) | 1 1 1 2 2 4 2 5 4 7 https://oeis.org/A241814
#
# in light of this, let's compute shortest path lengths as we go instead of
# precomputing them all
# test for regular graph (all degrees must be equal)
if not nx.is_regular(G) or not nx.is_connected(G):
raise nx.NetworkXError("Graph is not distance regular.")
path_length = defaultdict(dict)
bint = {} # 'b' intersection array
cint = {} # 'c' intersection array
# see https://doi.org/10.1016/j.ejc.2004.07.004, Theorem 1.5, page 81:
# the diameter of a distance-regular graph is at most (8 log_2 n) / 3,
# so let's compute it as we go in the hope that we can stop early
diam = 0
max_diameter_for_dr_graphs = (8 * log(len(G), 2)) / 3
for u, v in combinations_with_replacement(G, 2):
# compute needed shortest path lengths
pl_u = path_length[u]
if v not in pl_u:
pl_u.update(nx.single_source_shortest_path_length(G, u))
for x, distance in pl_u.items():
path_length[x][u] = distance
i = path_length[u][v]
diam = max(diam, i)
# diameter too large: graph can't be distance-regular
if diam > max_diameter_for_dr_graphs:
raise nx.NetworkXError("Graph is not distance regular.")
vnbrs = G[v]
# compute needed path lengths
for n in vnbrs:
pl_n = path_length[n]
if u not in pl_n:
pl_n.update(nx.single_source_shortest_path_length(G, n))
for x, distance in pl_n.items():
path_length[x][n] = distance
# number of neighbors of v at a distance of i-1 from u
c = sum(1 for n in vnbrs if pl_u[n] == i - 1)
# number of neighbors of v at a distance of i+1 from u
b = sum(1 for n in vnbrs if pl_u[n] == i + 1)
# b, c are independent of u and v
if cint.get(i, c) != c or bint.get(i, b) != b:
raise nx.NetworkXError("Graph is not distance regular")
bint[i] = b
cint[i] = c
return (
[bint.get(j, 0) for j in range(diam)],
[cint.get(j + 1, 0) for j in range(diam)],
)
# TODO There is a definition for directed strongly regular graphs.
[docs]
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatchable
def is_strongly_regular(G):
"""Returns True if and only if the given graph is strongly
regular.
An undirected graph is *strongly regular* if
* it is regular,
* each pair of adjacent vertices has the same number of neighbors in
common,
* each pair of nonadjacent vertices has the same number of neighbors
in common.
Each strongly regular graph is a distance-regular graph.
Conversely, if a distance-regular graph has diameter two, then it is
a strongly regular graph. For more information on distance-regular
graphs, see :func:`is_distance_regular`.
Parameters
----------
G : NetworkX graph
An undirected graph.
Returns
-------
bool
Whether `G` is strongly regular.
Examples
--------
The cycle graph on five vertices is strongly regular. It is
two-regular, each pair of adjacent vertices has no shared neighbors,
and each pair of nonadjacent vertices has one shared neighbor::
>>> G = nx.cycle_graph(5)
>>> nx.is_strongly_regular(G)
True
"""
# Here is an alternate implementation based directly on the
# definition of strongly regular graphs:
#
# return (all_equal(G.degree().values())
# and all_equal(len(common_neighbors(G, u, v))
# for u, v in G.edges())
# and all_equal(len(common_neighbors(G, u, v))
# for u, v in non_edges(G)))
#
# We instead use the fact that a distance-regular graph of diameter
# two is strongly regular.
return is_distance_regular(G) and diameter(G) == 2
