\MhGdZddlmZmZddlmZddlmZmZddl m Z ddl m Z ddl mZddlmZdd lZdd lmZmZmZgd ZGd d eZedejdd d+dZejd d dd d,dZejddd-dZeeeedZedejdd d.dZedejdd d.dZejddd/dZ ejdd d0d Z!ejddd/d!Z"ed"ejdd#d1dd d$d%Z#Gd&d'Z$ejd(d d d)d*Z%d S)2z= Algorithms for calculating min/max spanning trees/forests. ) dataclassfield)Enum)heappopheappush)count)isnan) itemgetter) PriorityQueueN) UnionFindnot_implemented_forpy_random_state) minimum_spanning_edgesmaximum_spanning_edgesminimum_spanning_treemaximum_spanning_treenumber_of_spanning_treesrandom_spanning_treepartition_spanning_tree EdgePartitionSpanningTreeIteratorceZdZdZdZdZdZdS)rz An enum to store the state of an edge partition. The enum is written to the edges of a graph before being pasted to `kruskal_mst_edges`. Options are: - EdgePartition.OPEN - EdgePartition.INCLUDED - EdgePartition.EXCLUDED rN)__name__ __module__ __qualname____doc__OPENINCLUDEDEXCLUDED\/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/networkx/algorithms/tree/mst.pyrrs) DHHHHr#r multigraphweightdata) edge_attrspreserve_edge_attrsTFc# Kt}fd fd|D}d|D}|rq fd|D}d|D}|D]>\}} } |||| kr&|r|| | fVn|| fV||| ?|odSdS)uIterate over edges of a Borůvka's algorithm min/max spanning tree. Parameters ---------- G : NetworkX Graph The edges of `G` must have distinct weights, otherwise the edges may not form a tree. minimum : bool (default: True) Find the minimum (True) or maximum (False) spanning tree. weight : string (default: 'weight') The name of the edge attribute holding the edge weights. keys : bool (default: True) This argument is ignored since this function is not implemented for multigraphs; it exists only for consistency with the other minimum spanning tree functions. data : bool (default: True) Flag for whether to yield edge attribute dicts. If True, yield edges `(u, v, d)`, where `d` is the attribute dict. If False, yield edges `(u, v)`. ignore_nan : bool (default: False) If a NaN is found as an edge weight normally an exception is raised. If `ignore_nan is True` then that edge is ignored instead. c rdnd}td}d}tj|dD]Q}|d d|z}t |rr3d|}t |||kr|}|}R|S)zReturns the optimum (minimum or maximum) edge on the edge boundary of the given set of nodes. A return value of ``None`` indicates an empty boundary. rinfNTr'"NaN found as an edge weight. Edge )floatnx edge_boundarygetr ValueError) componentsignminwtboundaryewtmsgG ignore_nanminimumr&s r$ best_edgez$boruvka_mst_edges..best_edgeSs#qqe !!YT:::  A261%%,BRyy &>1>> oo%Ezzr#c3.K|]}|VdSNr".0r5r?s r$ z$boruvka_mst_edges..ks-II9))I&&IIIIIIr#cg|]}||SrAr"rCedges r$ z%boruvka_mst_edges..lsBBB41A$1A1A1Ar#c3.K|]}|VdSrAr"rBs r$rDz$boruvka_mst_edges..s-MMyii **MMMMMMr#cg|]}||SrAr"rFs r$rHz%boruvka_mst_edges..sFFFtT5Ed5E5E5Er#N)r to_setsunion) r<r>r&keysr'r=forest best_edgesuvdr?s ``` ` @r$boruvka_mst_edgesrS-s@Hq\\F0JIII8H8HIIIJBB:BBBJ # NMMMFNNr&rMr'r=rTsubtreesrZincluded_edges open_edgesr9rRr:rGsorted_open_edges sorted_edgesrPrQks r$kruskal_mst_edgesrds\{{H#T--T""NJ $$ bE UU61   99 G E!EEFF Fuqy 55  }5 5 5  ! !$ ' ' ' ' UU9  !7 7 7    d # # # #P"::a==AAA"::a==$OOO+,,,!L%~ %* % %NB1a{hqk)) #&Aqj((((Ag #Ag d q!$$$ % %( % %KB1a{hqk))Q'MMMMQ$JJJq!$$$  % %r#c #K|}t|}t}|rdnd} |r|} g} | h} |r|j| D]\} }|D]k\}}||d| z}t|r|r0d| | ||f}t|t| |t|| | ||fln|j| D]i\} }||d| z}t|r|r0d| | |f}t|t| |t|| | |fj|r| r|rt| \}}} } }}nt| \}}} } }| | vs| |vr=|r|r|r | | ||fVn| | |fVn|r| | |fVn| | fV| | | | |r|j| D]\}}|| vr |D]k\}}||d| z}t|r|r0d| |||f}t|t| |t|| |||fln|j| D]n\}}|| vr ||d| z}t|r|r5d| ||f}t|t| |t|| ||fo|r| |dSdS)aIterate over edges of Prim's algorithm min/max spanning tree. Parameters ---------- G : NetworkX Graph The graph holding the tree of interest. minimum : bool (default: True) Find the minimum (True) or maximum (False) spanning tree. weight : string (default: 'weight') The name of the edge attribute holding the edge weights. keys : bool (default: True) If `G` is a multigraph, `keys` controls whether edge keys ar yielded. Otherwise `keys` is ignored. data : bool (default: True) Flag for whether to yield edge attribute dicts. If True, yield edges `(u, v, d)`, where `d` is the attribute dict. If False, yield edges `(u, v)`. ignore_nan : bool (default: False) If a NaN is found as an edge weight normally an exception is raised. If `ignore_nan is True` then that edge is ignored instead. rr,r/N)rYsetrpopadjitemsr3r r4rnextradddiscard)r<r>r&rMr'r=rYnodescr6rPfrontiervisitedrQkeydictrcrRr:r;W_wk2d2 new_weights r$prim_mst_edgesrxsl:OO%%M FFE A 11RD BH IIKK#  ;eAhnn.. B B 7#MMOOBBDAqvq))D0BRyy.%%$QAq!Q<QQ(oo-XDGGQ1a'@AAAAB Ba(( ; ;1UU61%%,99*!! J1ayJJC$S//)BQAq#9::::+ H+ H 2#*8#4#4 1aAqq ' 1 1 1aAG||q~~  "Q1*$$$$Q'MMMMQ'MMMMQ$JJJ KKNNN MM!    H"#%(.."2"2 P PJAwG|| ")--//PPB%'VVFA%6%6%=  ,,2)) ("W1bRT~"W"WC",S//1 JQAr2+NOOOOP PU1X^^-- H HEArG|| !#!2!2T!9JZ((.%%$OAq":OO(oo-X DGGQ2'FGGGGW+ H+ H/ BHBHBHBHBHr#)boruvkauborůvkakruskalprimdirectedrzc t|}n'#t$r}|d}t||d}~wwxYw||d||||S)u Generate edges in a minimum spanning forest of an undirected weighted graph. A minimum spanning tree is a subgraph of the graph (a tree) with the minimum sum of edge weights. A spanning forest is a union of the spanning trees for each connected component of the graph. Parameters ---------- G : undirected Graph An undirected graph. If `G` is connected, then the algorithm finds a spanning tree. Otherwise, a spanning forest is found. algorithm : string The algorithm to use when finding a minimum spanning tree. Valid choices are 'kruskal', 'prim', or 'boruvka'. The default is 'kruskal'. weight : string Edge data key to use for weight (default 'weight'). keys : bool Whether to yield edge key in multigraphs in addition to the edge. If `G` is not a multigraph, this is ignored. data : bool, optional If True yield the edge data along with the edge. ignore_nan : bool (default: False) If a NaN is found as an edge weight normally an exception is raised. If `ignore_nan is True` then that edge is ignored instead. Returns ------- edges : iterator An iterator over edges in a maximum spanning tree of `G`. Edges connecting nodes `u` and `v` are represented as tuples: `(u, v, k, d)` or `(u, v, k)` or `(u, v, d)` or `(u, v)` If `G` is a multigraph, `keys` indicates whether the edge key `k` will be reported in the third position in the edge tuple. `data` indicates whether the edge datadict `d` will appear at the end of the edge tuple. If `G` is not a multigraph, the tuples are `(u, v, d)` if `data` is True or `(u, v)` if `data` is False. Examples -------- >>> from networkx.algorithms import tree Find minimum spanning edges by Kruskal's algorithm >>> G = nx.cycle_graph(4) >>> G.add_edge(0, 3, weight=2) >>> mst = tree.minimum_spanning_edges(G, algorithm="kruskal", data=False) >>> edgelist = list(mst) >>> sorted(sorted(e) for e in edgelist) [[0, 1], [1, 2], [2, 3]] Find minimum spanning edges by Prim's algorithm >>> G = nx.cycle_graph(4) >>> G.add_edge(0, 3, weight=2) >>> mst = tree.minimum_spanning_edges(G, algorithm="prim", data=False) >>> edgelist = list(mst) >>> sorted(sorted(e) for e in edgelist) [[0, 1], [1, 2], [2, 3]] Notes ----- For Borůvka's algorithm, each edge must have a weight attribute, and each edge weight must be distinct. For the other algorithms, if the graph edges do not have a weight attribute a default weight of 1 will be used. Modified code from David Eppstein, April 2006 http://www.ics.uci.edu/~eppstein/PADS/ ( is not a valid choice for an algorithm.NTr>r&rMr'r= ALGORITHMSKeyErrorr4 r< algorithmr&rMr'r=algoerrr;s r$rrqswh')$ '''DDDoo3&' 4 4T     4/4c t|}n'#t$r}|d}t||d}~wwxYw||d||||S)u Generate edges in a maximum spanning forest of an undirected weighted graph. A maximum spanning tree is a subgraph of the graph (a tree) with the maximum possible sum of edge weights. A spanning forest is a union of the spanning trees for each connected component of the graph. Parameters ---------- G : undirected Graph An undirected graph. If `G` is connected, then the algorithm finds a spanning tree. Otherwise, a spanning forest is found. algorithm : string The algorithm to use when finding a maximum spanning tree. Valid choices are 'kruskal', 'prim', or 'boruvka'. The default is 'kruskal'. weight : string Edge data key to use for weight (default 'weight'). keys : bool Whether to yield edge key in multigraphs in addition to the edge. If `G` is not a multigraph, this is ignored. data : bool, optional If True yield the edge data along with the edge. ignore_nan : bool (default: False) If a NaN is found as an edge weight normally an exception is raised. If `ignore_nan is True` then that edge is ignored instead. Returns ------- edges : iterator An iterator over edges in a maximum spanning tree of `G`. Edges connecting nodes `u` and `v` are represented as tuples: `(u, v, k, d)` or `(u, v, k)` or `(u, v, d)` or `(u, v)` If `G` is a multigraph, `keys` indicates whether the edge key `k` will be reported in the third position in the edge tuple. `data` indicates whether the edge datadict `d` will appear at the end of the edge tuple. If `G` is not a multigraph, the tuples are `(u, v, d)` if `data` is True or `(u, v)` if `data` is False. Examples -------- >>> from networkx.algorithms import tree Find maximum spanning edges by Kruskal's algorithm >>> G = nx.cycle_graph(4) >>> G.add_edge(0, 3, weight=2) >>> mst = tree.maximum_spanning_edges(G, algorithm="kruskal", data=False) >>> edgelist = list(mst) >>> sorted(sorted(e) for e in edgelist) [[0, 1], [0, 3], [1, 2]] Find maximum spanning edges by Prim's algorithm >>> G = nx.cycle_graph(4) >>> G.add_edge(0, 3, weight=2) # assign weight 2 to edge 0-3 >>> mst = tree.maximum_spanning_edges(G, algorithm="prim", data=False) >>> edgelist = list(mst) >>> sorted(sorted(e) for e in edgelist) [[0, 1], [0, 3], [2, 3]] Notes ----- For Borůvka's algorithm, each edge must have a weight attribute, and each edge weight must be distinct. For the other algorithms, if the graph edges do not have a weight attribute a default weight of 1 will be used. Modified code from David Eppstein, April 2006 http://www.ics.uci.edu/~eppstein/PADS/ r~NFrrrs r$rrswf')$ '''DDDoo3&' 4 5d*   r)preserve_all_attrs returns_graphct|||dd|}|}|j|j||j|||S)uReturns a minimum spanning tree or forest on an undirected graph `G`. Parameters ---------- G : undirected graph An undirected graph. If `G` is connected, then the algorithm finds a spanning tree. Otherwise, a spanning forest is found. weight : str Data key to use for edge weights. algorithm : string The algorithm to use when finding a minimum spanning tree. Valid choices are 'kruskal', 'prim', or 'boruvka'. The default is 'kruskal'. ignore_nan : bool (default: False) If a NaN is found as an edge weight normally an exception is raised. If `ignore_nan is True` then that edge is ignored instead. Returns ------- G : NetworkX Graph A minimum spanning tree or forest. Examples -------- >>> G = nx.cycle_graph(4) >>> G.add_edge(0, 3, weight=2) >>> T = nx.minimum_spanning_tree(G) >>> sorted(T.edges(data=True)) [(0, 1, {}), (1, 2, {}), (2, 3, {})] Notes ----- For Borůvka's algorithm, each edge must have a weight attribute, and each edge weight must be distinct. For the other algorithms, if the graph edges do not have a weight attribute a default weight of 1 will be used. There may be more than one tree with the same minimum or maximum weight. See :mod:`networkx.tree.recognition` for more detailed definitions. Isolated nodes with self-loops are in the tree as edgeless isolated nodes. TrMr'r=)r __class__graphupdateadd_nodes_fromrmriadd_edges_fromr<r&rr=rZTs r$rr.sd # 9f4dz   E AGNN17QW]]__%%%U Hr#rTc t|||dd||}|}|j|j||j|||S)u Find a spanning tree while respecting a partition of edges. Edges can be flagged as either `INCLUDED` which are required to be in the returned tree, `EXCLUDED`, which cannot be in the returned tree and `OPEN`. This is used in the SpanningTreeIterator to create new partitions following the algorithm of Sörensen and Janssens [1]_. Parameters ---------- G : undirected graph An undirected graph. minimum : bool (default: True) Determines whether the returned tree is the minimum spanning tree of the partition of the maximum one. weight : str Data key to use for edge weights. partition : str The key for the edge attribute containing the partition data on the graph. Edges can be included, excluded or open using the `EdgePartition` enum. ignore_nan : bool (default: False) If a NaN is found as an edge weight normally an exception is raised. If `ignore_nan is True` then that edge is ignored instead. Returns ------- G : NetworkX Graph A minimum spanning tree using all of the included edges in the graph and none of the excluded edges. References ---------- .. [1] G.K. Janssens, K. Sörensen, An algorithm to generate all spanning trees in order of increasing cost, Pesquisa Operacional, 2005-08, Vol. 25 (2), p. 219-229, https://www.scielo.br/j/pope/a/XHswBwRwJyrfL88dmMwYNWp/?lang=en T)rMr'r=rT)rdrrrrrmrir)r<r>r&rTr=rZrs r$rrjs`       E AGNN17QW]]__%%%U Hr#c6t|||dd|}t|}|}|j|j||j|||S)uReturns a maximum spanning tree or forest on an undirected graph `G`. Parameters ---------- G : undirected graph An undirected graph. If `G` is connected, then the algorithm finds a spanning tree. Otherwise, a spanning forest is found. weight : str Data key to use for edge weights. algorithm : string The algorithm to use when finding a maximum spanning tree. Valid choices are 'kruskal', 'prim', or 'boruvka'. The default is 'kruskal'. ignore_nan : bool (default: False) If a NaN is found as an edge weight normally an exception is raised. If `ignore_nan is True` then that edge is ignored instead. Returns ------- G : NetworkX Graph A maximum spanning tree or forest. Examples -------- >>> G = nx.cycle_graph(4) >>> G.add_edge(0, 3, weight=2) >>> T = nx.maximum_spanning_tree(G) >>> sorted(T.edges(data=True)) [(0, 1, {}), (0, 3, {'weight': 2}), (1, 2, {})] Notes ----- For Borůvka's algorithm, each edge must have a weight attribute, and each edge weight must be distinct. For the other algorithms, if the graph edges do not have a weight attribute a default weight of 1 will be used. There may be more than one tree with the same minimum or maximum weight. See :mod:`networkx.tree.recognition` for more detailed definitions. Isolated nodes with self-loops are in the tree as edgeless isolated nodes. Tr) rlistrrrrrmrirrs r$rrsh # 9f4dz   E KKE AGNN17QW]]__%%%U Hr#)r)r)multiplicativeseedczfdfd}fd}dkrtjjSt d}t t }|||D]i\}} ||| |nd} |\} } || |} | || | f}|| jvrktj| |d }|||}r | |z| z }n?|| ztj |z|z}|tj | z| z}||z }nd }| d d }||kr || fn|| z } || ftdz kr,tj}||cSkt!d td )uc Sample a random spanning tree using the edges weights of `G`. This function supports two different methods for determining the probability of the graph. If ``multiplicative=True``, the probability is based on the product of edge weights, and if ``multiplicative=False`` it is based on the sum of the edge weight. However, since it is easier to determine the total weight of all spanning trees for the multiplicative version, that is significantly faster and should be used if possible. Additionally, setting `weight` to `None` will cause a spanning tree to be selected with uniform probability. The function uses algorithm A8 in [1]_ . Parameters ---------- G : nx.Graph An undirected version of the original graph. weight : string The edge key for the edge attribute holding edge weight. multiplicative : bool, default=True If `True`, the probability of each tree is the product of its edge weight over the sum of the product of all the spanning trees in the graph. If `False`, the probability is the sum of its edge weight over the sum of the sum of weights for all spanning trees in the graph. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness`. Returns ------- nx.Graph A spanning tree using the distribution defined by the weight of the tree. References ---------- .. [1] V. Kulkarni, Generating random combinatorial objects, Journal of Algorithms, 11 (1990), pp. 185–207 cB||vr|S|||}|||<|S)a We can think of clusters of contracted nodes as having one representative in the graph. Each node which is not in merged_nodes is still its own representative. Since a representative can be later contracted, we need to recursively search though the dict to find the final representative, but once we know it we can use path compression to speed up the access of the representative for next time. This cannot be replaced by the standard NetworkX union_find since that data structure will merge nodes with less representing nodes into the one with more representing nodes but this function requires we merge them using the order that contract_edges contracts using. Parameters ---------- merged_nodes : dict The dict storing the mapping from node to representative node The node whose representative we seek Returns ------- The representative of the `node` r") merged_nodesnoderep find_nodes r$rz'random_spanning_tree..find_nodes:2 | # #K)L,t*<==C!$L Jr#cVtj}t| }||i}D]B\}} ||} ||}||kr$tj|||dd|||<C||fS)a For the graph `G`, remove all edges not in the set `V` and then contract all edges in the set `U`. Returns ------- A copy of `G` which has had all edges not in `V` removed and all edges in `U` contracted. )incoming_graph_dataF) self_loopscopy)r1 MultiGraphrfrZ differenceremove_edges_fromcontracted_nodes) resultedges_to_removerrPrQu_repv_repr<UVrs r$ prepare_graphz+random_spanning_tree..prepare_graph7s1555fllnn--88;;  111  ( (DAqIlA..EIlA..E~~  uU S S S S"'L  V##r#c rt||S|dkr@||dSd}||D]9\}}}||t jt j|||fddzz }:|S) a^ Find the sum of weights of the spanning trees of `G` using the appropriate `method`. This is easy if the chosen method is 'multiplicative', since we can use Kirchhoff's Tree Matrix Theorem directly. However, with the 'additive' method, this process is slightly more complex and less computationally efficient as we have to find the number of spanning trees which contain each possible edge in the graph. Parameters ---------- G : NetworkX Graph The graph to find the total weight of all spanning trees on. weight : string The key for the weight edge attribute of the graph. Returns ------- float The sum of either the multiplicative or additive weight for all spanning trees in the graph. r&rr.rrFrGrN)rnumber_of_edgesrZ__iter____next__r1contracted_edge)r<r&totalrPrQrtrs r$spanning_tree_total_weightz8random_spanning_tree..spanning_tree_total_weightbs2  +Af=== =   ""a''wwFw++4466??AA!DD wwFw33GAq!Q!<*1Aq6eLLL#"""EE r#rrNrFrgg?zSomething went wrong! Only z edges in the spanning tree!)number_of_nodesr1 empty_graphrmrfrZrshufflerruniformremoverklenGraphr Exception)r<r&rrrrst_cached_valueshuffled_edgesrPrQe_weightnode_map prepared_GG_total_tree_weightrep_edge prepared_G_eG_e_total_tree_weight threshold numerator denominatorz spanning_treerrrs` ` @@@r$rrs\@)$)$)$)$)$)$)$)$V22222h Q~ag&&& AO AGGIIA!''))__NLL   *!*!1&,&81Q476??a,}*88VLL Ih**IIh,B,BC z' ' '-eL%?$>|V$T$T ! 4$'<r= partition_key)rr<r&r>r=s r$__init__zSpanningTreeIterator.__init__sF($(!  $ J r#cZt|_||jt |j|j|j|j|j |j}|j | |jr|n| i|S)zu Returns ------- SpanningTreeIterator The iterator object for this graph r) r partition_queue_clear_partitionr<rr>r&rr=sizeputr)rrs r$rzSpanningTreeIterator.__iter__s - df%%%, FDL$+t/A4?  $dk$ " "    NNF::J; K K    r#cX|jr |`|`t|j}||t |j|j|j|j |j }| ||| ||S)z Returns ------- (multi)Graph The spanning tree of next greatest weight, which ties broken arbitrarily. ) remptyr< StopIterationr3_write_partitionrr>r&rr= _partitionr)rrT next_trees r$rzSpanningTreeIterator.__next__"s   % % ' ' , (,,..  i(((+ FDL$+t/A4?    9--- i(((r#c|d|j}|d|j}|jD]}||jvrtj|j|<tj|j|<||t|j |j |j |j |j }||j }tj|r=|j r|n| |_|j||j|_dS)a| Create new partitions based of the minimum spanning tree of the current minimum partition. Parameters ---------- partition : Partition The Partition instance used to generate the current minimum spanning tree. partition_tree : nx.Graph The minimum spanning tree of the input partition. rrN)rrrrZrr!r rrr<r>r&rr=rr1 is_connectedrrrr)rrTpartition_treep1p2r9p1_mst p1_mst_weights r$rzSpanningTreeIterator._partition8sC^^Ay7<<>> ? ? ^^Ay7<<>> ? ?% = =A 000'4'=!!$'4'=!!$%%b)))0FLK&O !' 4; ? ? ?6**<59\$UMM ~BM(,,R[[]];;;$&$5$:$:$<$<!' = =r#c"|j}|j}|j}|r|ddn|d}|D]5^}}|t |tj||<6dS)a Writes the desired partition into the graph to calculate the minimum spanning tree. Parameters ---------- partition : Partition A Partition dataclass describing a partition on the edges of the graph. TrVr.N) rrr<rYrZr3tuplerr)rrTrrr<rZr9rRs r$rz%SpanningTreeIterator._write_partition]s#1* F./__->-> VAGGDG ) ) )AGGQUGDVDV  P PEQ-11%((M-> VAGGDG ) ) )AGGQUGDVDV  % %EQ!!m$ % %r#N)r&TF) rrrrrrrrrrrrr"r#r$rrs&YT            <&,#=#=#=JPPP, % % % % %r#r)r()rootr&cddl}t|dkrtjdtj|sqtj|sdStj||}t|j |ddddfStj d|vrtj dtj |sdSgfd|Dz}tj ||| }||d }||z }t|j |ddddfS) ugReturns the number of spanning trees in `G`. A spanning tree for an undirected graph is a tree that connects all nodes in the graph. For a directed graph, the analog of a spanning tree is called a (spanning) arborescence. The arborescence includes a unique directed path from the `root` node to each other node. The graph must be weakly connected, and the root must be a node that includes all nodes as successors [3]_. Note that to avoid discussing sink-roots and reverse-arborescences, we have reversed the edge orientation from [3]_ and use the in-degree laplacian. This function (when `weight` is `None`) returns the number of spanning trees for an undirected graph and the number of arborescences from a single root node for a directed graph. When `weight` is the name of an edge attribute which holds the weight value of each edge, the function returns the sum over all trees of the multiplicative weight of each tree. That is, the weight of the tree is the product of its edge weights. Kirchoff's Tree Matrix Theorem states that any cofactor of the Laplacian matrix of a graph is the number of spanning trees in the graph. (Here we use cofactors for a diagonal entry so that the cofactor becomes the determinant of the matrix with one row and its matching column removed.) For a weighted Laplacian matrix, the cofactor is the sum across all spanning trees of the multiplicative weight of each tree. That is, the weight of each tree is the product of its edge weights. The theorem is also known as Kirchhoff's theorem [1]_ and the Matrix-Tree theorem [2]_. For directed graphs, a similar theorem (Tutte's Theorem) holds with the cofactor chosen to be the one with row and column removed that correspond to the root. The cofactor is the number of arborescences with the specified node as root. And the weighted version gives the sum of the arborescence weights with root `root`. The arborescence weight is the product of its edge weights. Parameters ---------- G : NetworkX graph root : node A node in the directed graph `G` that has all nodes as descendants. (This is ignored for undirected graphs.) weight : string or None, optional (default=None) The name of the edge attribute holding the edge weight. If `None`, then each edge is assumed to have a weight of 1. Returns ------- Number Undirected graphs: The number of spanning trees of the graph `G`. Or the sum of all spanning tree weights of the graph `G` where the weight of a tree is the product of its edge weights. Directed graphs: The number of arborescences of `G` rooted at node `root`. Or the sum of all arborescence weights of the graph `G` with specified root where the weight of an arborescence is the product of its edge weights. Raises ------ NetworkXPointlessConcept If `G` does not contain any nodes. NetworkXError If the graph `G` is directed and the root node is not specified or is not in G. Examples -------- >>> G = nx.complete_graph(5) >>> round(nx.number_of_spanning_trees(G)) 125 >>> G = nx.Graph() >>> G.add_edge(1, 2, weight=2) >>> G.add_edge(1, 3, weight=1) >>> G.add_edge(2, 3, weight=1) >>> round(nx.number_of_spanning_trees(G, weight="weight")) 5 Notes ----- Self-loops are excluded. Multi-edges are contracted in one edge equal to the sum of the weights. References ---------- .. [1] Wikipedia "Kirchhoff's theorem." https://en.wikipedia.org/wiki/Kirchhoff%27s_theorem .. [2] Kirchhoff, G. R. Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung Galvanischer Ströme geführt wird Annalen der Physik und Chemie, vol. 72, pp. 497-508, 1847. .. [3] Margoliash, J. 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