9i SrSSKJr SSKJr SSKJr SSKrSSK J r SSK J r /S Qr \RS 5rS r\ "S 5\ "S 5\RS555r\ "S 5\ "S 5\RS555rg)zI ======================= Distance-regular graphs ======================= ) defaultdict)combinations_with_replacement)logN)not_implemented_for)diameter)is_distance_regularis_strongly_regularintersection_arrayglobal_parameterscP[U5 g![Ra gf=f)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 d/var/www/html/land-doc-ocr/venv/lib/python3.13/site-packages/networkx/algorithms/distance_regular.pyr r s+R1  s %%c@^U4Sj[TS/-S/U-55$)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 c3D># UHupUTSU- U- U4v M g7f)rN).0xybs r $global_parameters..qs( C-BTQQ!q1 a -Bs r)zip)rcs` rr r Hs%R DSaS1#'-B CCdirected multigraphc^^[R"U5(a"U5(d[R"S5e[ [ 5n0n0nSnS[ [U5S5-S- n[US5GHVupgXmUT;aETR[R"X55 TR5H upXUU'M XUm[UT5nXE:a[R"S5eXn U HSn Xn Xl;dMU R[R"X 55 U R5H upXUU 'M MU [UU4SjU 55n [UU4SjU 55nURTU 5U :wdURTU5U:wa[R"S5eXT'XT'GMY [U5Vs/sHoRUS5PM sn[U5Vs/sHoRUS -S5PM sn4$s snfs snf) 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.rc3D># UHnTUTS- :XdMSv M g7frNrrnipl_us rr%intersection_array.."55aDGq1u$45  c3D># UHnTUTS-:XdMSv M g7fr&rr's rrr+r,r-zGraph is not distance regularr)r is_regular is_connectedrrdictrlenrupdate"single_source_shortest_path_lengthitemsmaxsumgetrange)r path_lengthbintcintdiammax_diameter_for_dr_graphsuvrdistancevnbrsr(pl_nrrjr)r*s @@rr r tsd ==  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 $((1a.A"5""#BC CQQA4F"'t-A!Q-%*4[1[!a% [1 -1s 3H?IcB[U5=(a [U5S:H$)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 sj q ! 6hqkQ&66r)__doc__ collectionsr itertoolsrmathrnetworkxrnetworkx.utilsrdistance_measuresr__all__ _dispatchabler r r r rrrrOs $3.' ,,^)DXZ \"`#!`HZ \"27#!27r