# HG changeset patch
# User Nathann Cohen <nathann.cohen@gmail.com>
# Date 1269341571 3600
# Node ID f17430f5f6f16cd7968ddd265b53d545d69408f0
# Parent 9de022253f37a53a8319d6964e4113db7988ac21
#8513: small fixes in the docstrings of generic graph and undirected graph classes
diff r 9de022253f37 r f17430f5f6f1 sage/graphs/generic_graph.py
a

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2916  2916  itself has no cut vertices. Two distinct blocks cannot overlap in 
2917  2917  more than a single cut vertex. 
2918  2918  
2919   OUTPUT: ( B, C ), where B is a list of blocks each is a list of 
2920   vertices and the blocks are the corresponding induced subgraphs 
2921   and C is a list of cut vertices. 
 2919  OUTPUT: ``( B, C )``, where ``B`` is a list of blocks each is 
 2920  a list of vertices and the blocks are the corresponding induced 
 2921  subgraphsand ``C`` is a list of cut vertices. 
2922  2922  
2923  2923  EXAMPLES:: 
2924  2924  
… 
… 

2949  2949  ... 
2950  2950  NotImplementedError: ... 
2951  2951  
2952   ALGORITHM: 8.3.8 in [1]. Notice that the termination condition on 
2953   line (23) of the algorithm uses "p[v] == 0" which in the book 
2954   means that the parent is undefined; in this case, v must be the 
2955   root s. Since our vertex names start with 0, we substitute instead 
2956   the condition "v == s". This is the terminating condition used 
 2952  ALGORITHM: 8.3.8 in [Jungnickel05]_. Notice that the termination condition on 
 2953  line (23) of the algorithm uses ``p[v] == 0`` which in the book 
 2954  means that the parent is undefined; in this case, `v` must be the 
 2955  root `s`. Since our vertex names start with `0`, we substitute instead 
 2956  the condition ``v == s``. This is the terminating condition used 
2957  2957  in the general Depth First Search tree in Algorithm 8.2.1. 
2958  2958  
2959  2959  REFERENCE: 
2960  2960  
2961    [1] D. Jungnickel, Graphs, Networks and Algorithms, 
 2961  .. [Jungnickel05] D. Jungnickel, Graphs, Networks and Algorithms, 
2962  2962  Springer, 2005. 
2963  2963  """ 
2964  2964  if not self: # empty graph 
… 
… 

3826  3826  
3827  3827  This function returns a list of such paths. 
3828  3828  
3829   NOTE: 
3830   
3831   This function is topological : it does not take the eventual 
3832   weights of the edges into account. 
 3829  .. NOTE:: 
 3830  
 3831  This function is topological : it does not take the eventual 
 3832  weights of the edges into account. 
3833  3833  
3834  3834  EXAMPLE: 
3835  3835  
… 
… 

6583  6583  ordered list. 
6584  6584  
6585  6585  The clustering coefficient of a graph is the fraction of possible 
6586   triangles that are triangles, c_i = triangles_i / 
6587   (k_i\*(k_i1)/2) where k_i is the degree of vertex i, [1]. A 
6588   coefficient for the whole graph is the average of the c_i. 
 6586  triangles that are triangles, `c_i = triangles_i / 
 6587  (k_i\*(k_i1)/2)` where `k_i` is the degree of vertex `i`, [1]. A 
 6588  coefficient for the whole graph is the average of the `c_i`. 
6589  6589  Transitivity is the fraction of all possible triangles which are 
6590  6590  triangles, T = 3\*triangles/triads, [HSSNX]_. 
6591  6591  
… 
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6622  6622  Returns the average clustering coefficient. 
6623  6623  
6624  6624  The clustering coefficient of a graph is the fraction of possible 
6625   triangles that are triangles, c_i = triangles_i / 
6626   (k_i\*(k_i1)/2) where k_i is the degree of vertex i, [1]. A 
6627   coefficient for the whole graph is the average of the c_i. 
 6625  triangles that are triangles, `c_i = triangles_i / 
 6626  (k_i\*(k_i1)/2)` where `k_i` is the degree of vertex `i`, [1]. A 
 6627  coefficient for the whole graph is the average of the `c_i`. 
6628  6628  Transitivity is the fraction of all possible triangles which are 
6629  6629  triangles, T = 3\*triangles/triads, [1]. 
6630  6630  
… 
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6648  6648  ordered list. 
6649  6649  
6650  6650  The clustering coefficient of a graph is the fraction of possible 
6651   triangles that are triangles, c_i = triangles_i / 
6652   (k_i\*(k_i1)/2) where k_i is the degree of vertex i, [1]. A 
6653   coefficient for the whole graph is the average of the c_i. 
 6651  triangles that are triangles, `c_i = triangles_i / 
 6652  (k_i\*(k_i1)/2)` where `k_i` is the degree of vertex `i`, [1]. A 
 6653  coefficient for the whole graph is the average of the `c_i`. 
6654  6654  Transitivity is the fraction of all possible triangles which are 
6655  6655  triangles, T = 3\*triangles/triads, [1]. 
6656  6656  
… 
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6697  6697  graph. 
6698  6698  
6699  6699  The clustering coefficient of a graph is the fraction of possible 
6700   triangles that are triangles, c_i = triangles_i / 
6701   (k_i\*(k_i1)/2) where k_i is the degree of vertex i, [1]. A 
6702   coefficient for the whole graph is the average of the c_i. 
 6700  triangles that are triangles, `c_i = triangles_i / 
 6701  (k_i\*(k_i1)/2)` where `k_i` is the degree of vertex `i`, [1]. A 
 6702  coefficient for the whole graph is the average of the `c_i`. 
6703  6703  Transitivity is the fraction of all possible triangles which are 
6704  6704  triangles, T = 3\*triangles/triads, [1]. 
6705  6705  
6706  6706  REFERENCE: 
6707  6707  
6708    [1] Aric Hagberg, Dan Schult and Pieter Swart. NetworkX 
 6708  .. [1] Aric Hagberg, Dan Schult and Pieter Swart. NetworkX 
6709  6709  documentation. [Online] Available: 
6710  6710  https://networkx.lanl.gov/reference/networkx/ 
6711  6711  
… 
… 

8526  8526  """ 
8527  8527  Returns the Cartesian product of self and other. 
8528  8528  
8529   The Cartesian product of G and H is the graph L with vertex set 
8530   V(L) equal to the Cartesian product of the vertices V(G) and V(H), 
8531   and ((u,v), (w,x)) is an edge iff either  (u, w) is an edge of 
8532   self and v = x, or  (v, x) is an edge of other and u = w. 
 8529  The Cartesian product of `G` and `H` is the graph `L` with vertex set 
 8530  `V(L)` equal to the Cartesian product of the vertices `V(G)` and `V(H)`, 
 8531  and `((u,v), (w,x))` is an edge iff either  `(u, w)` is an edge of 
 8532  self and `v = x`, or  `(v, x)` is an edge of other and `u = w`. 
8533  8533  
8534  8534  EXAMPLES:: 
8535  8535  
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8573  8573  Returns the tensor product, also called the categorical product, of 
8574  8574  self and other. 
8575  8575  
8576   The tensor product of G and H is the graph L with vertex set V(L) 
8577   equal to the Cartesian product of the vertices V(G) and V(H), and 
8578   ((u,v), (w,x)) is an edge iff  (u, w) is an edge of self, and  
8579   (v, x) is an edge of other. 
 8576  The tensor product of `G` and `H` is the graph `L` with vertex set `V(L)` 
 8577  equal to the Cartesian product of the vertices `V(G)` and `V(H)`, and 
 8578  `((u,v), (w,x))` is an edge iff  `(u, w)` is an edge of self, and  
 8579  `(v, x)` is an edge of other. 
8580  8580  
8581  8581  EXAMPLES:: 
8582  8582  
diff r 9de022253f37 r f17430f5f6f1 sage/graphs/graph.py
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1411  1411  as a Graph object. 
1412  1412   When no solution exists, returns ``False`` 
1413  1413  
1414   NOTES: 
 1414  .. NOTE:: 
1415  1415  
1416    This algorithm computes the degreeconstrained subgraph of minimum weight. 
1417    If the graph's edges are weighted, these are taken into account. 
1418    This problem can be solved in polynomial time. 
 1416   This algorithm computes the degreeconstrained subgraph of minimum weight. 
 1417   If the graph's edges are weighted, these are taken into account. 
 1418   This problem can be solved in polynomial time. 
1419  1419  
1420  1420  EXAMPLES: 
1421  1421  
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1491  1491  
1492  1492  A digraph representing an orientation of the current graph. 
1493  1493  
1494   NOTES: 
 1494  .. NOTE:: 
1495  1495  
1496    This method assumes the graph is connected. 
1497    This algorithm works in O(m). 
 1496   This method assumes the graph is connected. 
 1497   This algorithm works in O(m). 
1498  1498  
1499  1499  EXAMPLE: 
1500  1500  
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2163  2163  by a list of vertices. A clique is an induced complete subgraph, and a 
2164  2164  maximal clique is one not contained in a larger one. 
2165  2165  
2166   NOTES: 
 2166  .. NOTE:: 
2167  2167  
2168    Currently only implemented for undirected graphs. Use to_undirected 
2169   to convert a digraph to an undirected graph. 
 2168  Currently only implemented for undirected graphs. Use to_undirected 
 2169  to convert a digraph to an undirected graph. 
2170  2170  
2171  2171  ALGORITHM: 
2172  2172  
… 
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2223  2223  by a list of vertices. A clique is an induced complete subgraph, and a 
2224  2224  maximum clique is one of maximal order. 
2225  2225  
2226   NOTES: 
 2226  .. NOTE:: 
2227  2227  
2228    Currently only implemented for undirected graphs. Use to_undirected 
2229   to convert a digraph to an undirected graph. 
 2228  Currently only implemented for undirected graphs. Use to_undirected 
 2229  to convert a digraph to an undirected graph. 
2230  2230  
2231  2231  ALGORITHM: 
2232  2232  
… 
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2262  2262  """ 
2263  2263  Returns the vertex set of a maximal order complete subgraph. 
2264  2264  
2265   NOTE: 
 2265  .. NOTE:: 
2266  2266  
2267    Currently only implemented for undirected graphs. Use to_undirected 
2268   to convert a digraph to an undirected graph. 
 2267  Currently only implemented for undirected graphs. Use to_undirected 
 2268  to convert a digraph to an undirected graph. 
2269  2269  
2270  2270  ALGORITHM: 
2271  2271  
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2289  2289  Returns the order of the largest clique of the graph (the clique 
2290  2290  number). 
2291  2291  
2292   NOTE: 
 2292  .. NOTE:: 
2293  2293  
2294    Currently only implemented for undirected graphs. Use ``to_undirected`` 
2295   to convert a digraph to an undirected graph. 
 2294  Currently only implemented for undirected graphs. Use ``to_undirected`` 
 2295  to convert a digraph to an undirected graph. 
2296  2296  
2297  2297  INPUT: 
2298  2298  
… 
… 

2334  2334  Returns a list of the number of maximal cliques containing each 
2335  2335  vertex. (Returns a single value if only one input vertex). 
2336  2336  
2337   NOTES: 
 2337  .. NOTE:: 
2338  2338  
2339    Currently only implemented for undirected graphs. Use to_undirected 
2340   to convert a digraph to an undirected graph. 
 2339  Currently only implemented for undirected graphs. Use to_undirected 
 2340  to convert a digraph to an undirected graph. 
2341  2341  
2342  2342  INPUT: 
2343  2343  
… 
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2387  2387  edges between maximal cliques with common members in the original 
2388  2388  graph. 
2389  2389  
2390   NOTES: 
 2390  .. NOTE:: 
2391  2391  
2392    Currently only implemented for undirected graphs. Use to_undirected 
2393   to convert a digraph to an undirected graph. 
 2392  Currently only implemented for undirected graphs. Use to_undirected 
 2393  to convert a digraph to an undirected graph. 
2394  2394  
2395  2395  INPUT: 
2396  2396  
… 
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2417  2417  graph. Right and left vertices are connected if the bottom vertex 
2418  2418  belongs to the clique represented by a top vertex. 
2419  2419  
2420   NOTES: 
 2420  .. NOTE:: 
2421  2421  
2422    Currently only implemented for undirected graphs. Use to_undirected 
2423   to convert a digraph to an undirected graph. 
 2422  Currently only implemented for undirected graphs. Use to_undirected 
 2423  to convert a digraph to an undirected graph. 
2424  2424  
2425  2425  EXAMPLES:: 
2426  2426  
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2442  2442  Returns a maximal independent set, which is a set of vertices which 
2443  2443  induces an empty subgraph. Uses Cliquer [NisOst2003]_. 
2444  2444  
2445   NOTES: 
 2445  .. NOTE:: 
2446  2446  
2447    Currently only implemented for undirected graphs. Use to_undirected 
2448   to convert a digraph to an undirected graph. 
 2447  Currently only implemented for undirected graphs. Use to_undirected 
 2448  to convert a digraph to an undirected graph. 
2449  2449  
2450  2450  EXAMPLES:: 
2451  2451  
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2462  2462  Returns a list of sizes of the largest maximal cliques containing 
2463  2463  each vertex. (Returns a single value if only one input vertex). 
2464  2464  
2465   NOTES: 
 2465  .. NOTE:: 
2466  2466  
2467    Currently only implemented for undirected graphs. Use to_undirected 
2468   to convert a digraph to an undirected graph. 
 2467  Currently only implemented for undirected graphs. Use to_undirected 
 2468  to convert a digraph to an undirected graph. 
2469  2469  
2470  2470  INPUT: 
2471  2471  
… 
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2535  2535  Returns the cliques containing each vertex, represented as a list 
2536  2536  of lists. (Returns a single list if only one input vertex). 
2537  2537  
2538   NOTE: 
 2538  .. NOTE:: 
2539  2539  
2540    Currently only implemented for undirected graphs. Use to_undirected 
2541   to convert a digraph to an undirected graph. 
 2540  Currently only implemented for undirected graphs. Use to_undirected 
 2541  to convert a digraph to an undirected graph. 
2542  2542  
2543  2543  INPUT: 
2544  2544  