## Mercurial > repos > shellac > sam_consensus_v3

### view env/lib/python3.9/site-packages/networkx/algorithms/distance_regular.py @ 0:4f3585e2f14b draft default tip

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author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |

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""" ======================= Distance-regular graphs ======================= """ 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", ] 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 def global_parameters(b, c): """Returns 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 : 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)) @not_implemented_for("directed", "multigraph") 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 """ # test for regular graph (all degrees must be equal) degree = iter(G.degree()) (_, k) = next(degree) for _, knext in degree: if knext != k: raise nx.NetworkXError("Graph is not distance regular.") k = knext path_length = dict(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 as e: # graph must be connected raise nx.NetworkXError("Graph is not distance regular.") from e # 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(j, 0) for j in range(diameter)], [cint.get(j + 1, 0) for j in range(diameter)], ) # TODO There is a definition for directed strongly regular graphs. @not_implemented_for("directed", "multigraph") 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