+ j  Rt^RIHt^RIHt^RIHt^RIt^RI H t ^RI H t .R Ot ]PR4tRt] !R 4] !R 4]PR 444t] !R 4] !R 4]PR 444tR#)zI ======================= Distance-regular graphs ======================= ) defaultdict)combinations_with_replacement)logN)not_implemented_for)diametercV\V4R# \PdR#i;i)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)intersection_arraynx NetworkXErrorGs&e/home/luca/screen-checking/.venv/lib/python3.14/site-packages/networkx/algorithms/distance_regular.pyis_distance_regularrs+R1  s ((cPaV3Rl\S^.,^.V,44#)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\<"TF!wrVS^,V, V, V3xK# R#5i)N).0xybs& r $global_parameters..qs( C-BTQQ!q1 a -Bs),)zip)rcsf&r global_parametersrHs%R DSaS1#'-B CCdirected multigraphcaa\P!V4'd\P!V4'g\P!R4h\ \ 4p/p/p^p^\ \V4^4,^, p\V^4EFwwrgW,oVS9dJSP\P!W44SP4FwrWV,V&K W,V,o\VS4pWE8d\P!R4hW,p V F\p W,p Wl9gKV P\P!W 44V P4FwrWV,V &K K^ \VV3RlV 44p \VV3RlV 44pVPSV 4V 8wgVPSV4V8wd\P!R4hWS&WS&EKz \V4Uu.uFqPV^4NK up\V4Uu.uFqPV^,^4NK up3#uupiuupi)a(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 zGraph is not distance regular.c3V<"TFpSV,S^, 8XgK^xK R#5iNrrnipl_us& r r%intersection_array.."55aDGq1u$45) )c3V<"TFpSV,S^,8XgK^xK R#5ir!rr#s& r rr'r(r)zGraph is not distance regular)r is_regular is_connectedr rdictrlenrupdate"single_source_shortest_path_lengthitemsmaxsumgetrange)r path_lengthbintcintdiammax_diameter_for_dr_graphsuvrdistancevnbrsr$pl_nrrjr%r&s& @@r rrtsd ==  2??1#5#5?@@d#K D D D"#c#a&!n"4!9-a3~ D= KK==aC D#zz| $,Aq! , N1 4|  ,""#CD DA>D} BAA!GH#'::Q $((1a.A"5""#BC CQQA4F"'t-A!Q-%*4[1[!a% [1 -1s I,!I1cD\V4;'d\V4^8H#)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 )rrr s&r is_strongly_regularrBs!j q ! 6 6hqkQ&66r)rrBrr)__doc__ collectionsr itertoolsrmathrnetworkxr networkx.utilsrdistance_measuresr__all__ _dispatchablerrrrBrrr rLs $3.' ,,^)DXZ \"`#!`HZ \"27#!27r