btj DdZddlmZddlmZddlmZddlZddl m Z ddl m Z gd Z ejd Zd Ze d e d ejdZe d e d ejdZdS)zI ======================= Distance-regular graphs ======================= ) defaultdict)combinations_with_replacement)logN)not_implemented_for)diameter)is_distance_regularis_strongly_regularintersection_arrayglobal_parameterscR t|dS#tj$rYdSwxYw)aReturns 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 TF)r nx NetworkXErrorGs b/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/networkx/algorithms/distance_regular.pyr r s@R1t uus &&cLfdtdgzdg|zDS)aReturns 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 c3@K|]\}}|d|z |z |fVdS)rN).0xybs r z$global_parameters..qs: C CTQQ!q1 a C C C C C Cr)zip)rcs` rr r Hs7R D C C CSaS1#'-B-B C C CCrdirected multigraphc tj|rtj|stjdt t }i id}dt t|dzdz }t|dD]\}}|||vrM tj || D]\}}||||<|||t|}||krtjd||}|D][} || } || vrM| tj || | D]\}}|||| <\tfd|D} tfd|D} | | ks | | krtjd| <| < fd t|Dfd t|DfS) aReturns 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 zGraph is not distance regular.rc3:K|]}|dz kdVdSrNrrnipl_us rrz%intersection_array..555aDGq1u$4$4$4$4$4$455rc3:K|]}|dzkdVdSr%rr&s rrz%intersection_array..r*rzGraph is not distance regularc<g|]}|dS)rget)rjbints r z&intersection_array..s%---A!Q---rcBg|]}|dzdS)rrr-)rr/cints rr1z&intersection_array..s+111!a%  111r)r is_regular is_connectedrrdictrlenrupdate"single_source_shortest_path_lengthitemsmaxsumr.range)r path_lengthdiammax_diameter_for_dr_graphsuvrdistancevnbrsr'pl_nrrr0r3r(r)s @@@@rr r tsd =  A2?1#5#5A?@@@d##K D D D"#c#a&&!nn"4!9-a33  11~ D== KK=aCC D D D#zz|| - - 8$, Aq!! N1 4|| , , ,"#CDD D! 1 1Aq>D}} BA!QGGHHH#'::<<11KAx(0KN1%% 555555555 5 5 555555555 5 5 88Aq>>Q  $((1a..A"5"5"#BCC CQQ .---t---1111U4[[111 rcFt|ot|dkS)aReturns 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 r")r rrs rr r s#j q ! ! 6hqkkQ&66r)__doc__ collectionsr itertoolsrmathrnetworkxrnetworkx.utilsrdistance_measuresr__all__ _dispatchabler r r r rrrrPs_ $#####333333......''''''   ,,,^)D)D)DXZ  \""``#"! `HZ  \""2727#"! 272727r