Source code for networkx.algorithms.distance_regular

"""
=======================
Distance-regular graphs
=======================
"""
#    Copyright (C) 2011 by 
#    Dheeraj M R <dheerajrav@gmail.com>
#    Aric Hagberg <aric.hagberg@gmail.com>
#    All rights reserved.
#    BSD license.
import networkx as nx
__author__ = """\n""".join(['Dheeraj M R <dheerajrav@gmail.com>',
                            'Aric Hagberg <aric.hagberg@gmail.com>'])

__all__ = ['is_distance_regular','intersection_array','global_parameters']

[docs]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: a=intersection_array(G) return True except nx.NetworkXError: return False
[docs]def global_parameters(b,c): """Return global parameters for a given intersection array. 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. 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,c: tuple of lists Returns ------- p : list of 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 """ d=len(b) ba=b[:] ca=c[:] ba.append(0) ca.insert(0,0) k = ba[0] aa = [k-x-y for x,y in zip(ba,ca)] return zip(*[ca,aa,ba])
[docs]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'sintersection 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 """ if G.is_multigraph() or G.is_directed(): raise nx.NetworkxException('Not implemented for directed ', 'or multiedge graphs.') # test for regular graph (all degrees must be equal) degree = G.degree_iter() (_,k) = next(degree) for _,knext in degree: if knext != k: raise nx.NetworkXError('Graph is not distance regular.') k = knext path_length = nx.all_pairs_shortest_path_length(G) diameter = max([max(path_length[n].values()) for n in path_length]) bint = {} # 'b' intersection array cint = {} # 'c' intersection array for u in G: for v in G: try: i = path_length[u][v] except KeyError: # graph must be connected raise nx.NetworkXError('Graph is not distance regular.') # number of neighbors of v at a distance of i-1 from u c = len([n for n in G[v] if path_length[n][u]==i-1]) # number of neighbors of v at a distance of i+1 from u b = len([n for n in G[v] if path_length[n][u]==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(i,0) for i in range(diameter)], [cint.get(i+1,0) for i in range(diameter)])