231 lines
6.7 KiB
Python
231 lines
6.7 KiB
Python
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"""
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=======================
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Distance-regular graphs
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=======================
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"""
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import networkx as nx
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from networkx.utils import not_implemented_for
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from .distance_measures import diameter
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__all__ = [
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"is_distance_regular",
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"is_strongly_regular",
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"intersection_array",
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"global_parameters",
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]
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def is_distance_regular(G):
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"""Returns True if the graph is distance regular, False otherwise.
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A connected graph G is distance-regular if for any nodes x,y
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and any integers i,j=0,1,...,d (where d is the graph
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diameter), the number of vertices at distance i from x and
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distance j from y depends only on i,j and the graph distance
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between x and y, independently of the choice of x and y.
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Parameters
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----------
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G: Networkx graph (undirected)
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Returns
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-------
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bool
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True if the graph is Distance Regular, False otherwise
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Examples
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--------
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>>> G = nx.hypercube_graph(6)
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>>> nx.is_distance_regular(G)
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True
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See Also
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--------
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intersection_array, global_parameters
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Notes
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-----
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For undirected and simple graphs only
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References
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----------
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.. [1] Brouwer, A. E.; Cohen, A. M.; and Neumaier, A.
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Distance-Regular Graphs. New York: Springer-Verlag, 1989.
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.. [2] Weisstein, Eric W. "Distance-Regular Graph."
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http://mathworld.wolfram.com/Distance-RegularGraph.html
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"""
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try:
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intersection_array(G)
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return True
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except nx.NetworkXError:
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return False
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def global_parameters(b, c):
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"""Returns global parameters for a given intersection array.
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Given a distance-regular graph G with integers b_i, c_i,i = 0,....,d
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such that for any 2 vertices x,y in G at a distance i=d(x,y), there
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are exactly c_i neighbors of y at a distance of i-1 from x and b_i
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neighbors of y at a distance of i+1 from x.
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Thus, a distance regular graph has the global parameters,
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[[c_0,a_0,b_0],[c_1,a_1,b_1],......,[c_d,a_d,b_d]] for the
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intersection array [b_0,b_1,.....b_{d-1};c_1,c_2,.....c_d]
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where a_i+b_i+c_i=k , k= degree of every vertex.
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Parameters
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----------
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b : list
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c : list
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Returns
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-------
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iterable
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An iterable over three tuples.
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Examples
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--------
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>>> G = nx.dodecahedral_graph()
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>>> b, c = nx.intersection_array(G)
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>>> list(nx.global_parameters(b, c))
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[(0, 0, 3), (1, 0, 2), (1, 1, 1), (1, 1, 1), (2, 0, 1), (3, 0, 0)]
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References
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----------
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.. [1] Weisstein, Eric W. "Global Parameters."
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From MathWorld--A Wolfram Web Resource.
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http://mathworld.wolfram.com/GlobalParameters.html
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See Also
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--------
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intersection_array
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"""
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return ((y, b[0] - x - y, x) for x, y in zip(b + [0], [0] + c))
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@not_implemented_for("directed", "multigraph")
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def intersection_array(G):
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"""Returns the intersection array of a distance-regular graph.
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Given a distance-regular graph G with integers b_i, c_i,i = 0,....,d
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such that for any 2 vertices x,y in G at a distance i=d(x,y), there
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are exactly c_i neighbors of y at a distance of i-1 from x and b_i
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neighbors of y at a distance of i+1 from x.
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A distance regular graph's intersection array is given by,
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[b_0,b_1,.....b_{d-1};c_1,c_2,.....c_d]
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Parameters
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----------
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G: Networkx graph (undirected)
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Returns
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-------
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b,c: tuple of lists
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Examples
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--------
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>>> G = nx.icosahedral_graph()
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>>> nx.intersection_array(G)
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([5, 2, 1], [1, 2, 5])
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References
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----------
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.. [1] Weisstein, Eric W. "Intersection Array."
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From MathWorld--A Wolfram Web Resource.
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http://mathworld.wolfram.com/IntersectionArray.html
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See Also
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--------
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global_parameters
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"""
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# test for regular graph (all degrees must be equal)
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degree = iter(G.degree())
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(_, k) = next(degree)
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for _, knext in degree:
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if knext != k:
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raise nx.NetworkXError("Graph is not distance regular.")
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k = knext
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path_length = dict(nx.all_pairs_shortest_path_length(G))
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diameter = max([max(path_length[n].values()) for n in path_length])
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bint = {} # 'b' intersection array
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cint = {} # 'c' intersection array
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for u in G:
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for v in G:
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try:
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i = path_length[u][v]
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except KeyError as e: # graph must be connected
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raise nx.NetworkXError("Graph is not distance regular.") from e
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# number of neighbors of v at a distance of i-1 from u
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c = len([n for n in G[v] if path_length[n][u] == i - 1])
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# number of neighbors of v at a distance of i+1 from u
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b = len([n for n in G[v] if path_length[n][u] == i + 1])
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# b,c are independent of u and v
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if cint.get(i, c) != c or bint.get(i, b) != b:
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raise nx.NetworkXError("Graph is not distance regular")
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bint[i] = b
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cint[i] = c
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return (
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[bint.get(j, 0) for j in range(diameter)],
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[cint.get(j + 1, 0) for j in range(diameter)],
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)
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# TODO There is a definition for directed strongly regular graphs.
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@not_implemented_for("directed", "multigraph")
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def is_strongly_regular(G):
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"""Returns True if and only if the given graph is strongly
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regular.
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An undirected graph is *strongly regular* if
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* it is regular,
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* each pair of adjacent vertices has the same number of neighbors in
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common,
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* each pair of nonadjacent vertices has the same number of neighbors
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in common.
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Each strongly regular graph is a distance-regular graph.
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Conversely, if a distance-regular graph has diameter two, then it is
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a strongly regular graph. For more information on distance-regular
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graphs, see :func:`is_distance_regular`.
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Parameters
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----------
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G : NetworkX graph
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An undirected graph.
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Returns
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-------
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bool
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Whether `G` is strongly regular.
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Examples
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--------
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The cycle graph on five vertices is strongly regular. It is
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two-regular, each pair of adjacent vertices has no shared neighbors,
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and each pair of nonadjacent vertices has one shared neighbor::
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>>> G = nx.cycle_graph(5)
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>>> nx.is_strongly_regular(G)
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True
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"""
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# Here is an alternate implementation based directly on the
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# definition of strongly regular graphs:
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#
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# return (all_equal(G.degree().values())
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# and all_equal(len(common_neighbors(G, u, v))
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# for u, v in G.edges())
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# and all_equal(len(common_neighbors(G, u, v))
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# for u, v in non_edges(G)))
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#
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# We instead use the fact that a distance-regular graph of diameter
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# two is strongly regular.
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return is_distance_regular(G) and diameter(G) == 2
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