Basic graphs - Graph Theory

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# Basic graphs

The methods defined here appear in `sage.graphs.graph_generators`.

sage.graphs.generators.basic.BullGraph(immutable=False)

  Return a bull graph with 5 nodes.

  A bull graph is named for its shape. It’s a triangle with horns. See the Wikipedia article Bull_graph for more information.

  INPUT:

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  PLOTTING:

  Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the bull graph is drawn as a triangle with the first node (0) on the bottom. The second and third nodes (1 and 2) complete the triangle. Node 3 is the horn connected to 1 and node 4 is the horn connected to node 2.

  EXAMPLES:

  Construct and show a bull graph:

  Sage

      sage: g = graphs.BullGraph(); g
      Bull graph: Graph on 5 vertices
      sage: g.show()                          # long time                             # needs sage.plot

  Python

      >>> from sage.all import *
      >>> g = graphs.BullGraph(); g
      Bull graph: Graph on 5 vertices
      >>> g.show()                          # long time                             # needs sage.plot

  The bull graph has 5 vertices and 5 edges. Its radius is 2, its diameter 3, and its girth 3. The bull graph is planar with chromatic number 3 and chromatic index also 3:

  Sage

      sage: g.order(); g.size()
      5
      5
      sage: g.radius(); g.diameter(); g.girth()
      2
      3
      3
      sage: g.chromatic_number()
      3

  [Python]

      >>> from sage.all import *
      >>> g.order(); g.size()
      5
      5
      >>> g.radius(); g.diameter(); g.girth()
      2
      3
      3
      >>> g.chromatic_number()
      3

  The bull graph has chromatic polynomial \(x(x - 2)(x - 1)^3\) and Tutte polynomial \(x^4 + x^3 + x^2 y\). Its characteristic polynomial is \(x(x^2 - x - 3)(x^2 + x - 1)\), which follows from the definition of characteristic polynomials for graphs, i.e. \(\det(xI - A)\), where \(x\) is a variable, \(A\) the adjacency matrix of the graph, and \(I\) the identity matrix of the same dimensions as \(A\):

  Sage

      sage: chrompoly = g.chromatic_polynomial()
      sage: x = chrompoly.parent()('x')
      sage: x * (x - 2) * (x - 1)^3 == chrompoly
      True

      sage: charpoly = g.characteristic_polynomial()
      sage: M = g.adjacency_matrix(); M
      [0 1 1 0 0]
      [1 0 1 1 0]
      [1 1 0 0 1]
      [0 1 0 0 0]
      [0 0 1 0 0]
      sage: Id = identity_matrix(ZZ, M.nrows())
      sage: D = x*Id - M
      sage: D.determinant() == charpoly                                               # needs sage.symbolic
      True
      sage: x * (x^2 - x - 3) * (x^2 + x - 1) == charpoly
      True

  Python

      >>> from sage.all import *
      >>> chrompoly = g.chromatic_polynomial()
      >>> x = chrompoly.parent()('x')
      >>> x * (x - Integer(2)) * (x - Integer(1))**Integer(3) == chrompoly
      True

      >>> charpoly = g.characteristic_polynomial()
      >>> M = g.adjacency_matrix(); M
      [0 1 1 0 0]
      [1 0 1 1 0]
      [1 1 0 0 1]
      [0 1 0 0 0]
      [0 0 1 0 0]
      >>> Id = identity_matrix(ZZ, M.nrows())
      >>> D = x*Id - M
      >>> D.determinant() == charpoly                                               # needs sage.symbolic
      True
      >>> x * (x**Integer(2) - x - Integer(3)) * (x**Integer(2) + x - Integer(1)) == charpoly
      True

sage.graphs.generators.basic.ButterflyGraph(immutable=False)

  Return the butterfly graph.

  Let \(C_3\) be the cycle graph on 3 vertices. The butterfly or bowtie graph is obtained by joining two copies of \(C_3\) at a common vertex, resulting in a graph that is isomorphic to the friendship graph \(F_2\). See the Wikipedia article Butterfly_graph for more information.

  See also

  - `FriendshipGraph()`

  INPUT:

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  EXAMPLES:

  The butterfly graph is a planar graph on 5 vertices and having 6 edges:

  Sage

      sage: G = graphs.ButterflyGraph(); G
      Butterfly graph: Graph on 5 vertices
      sage: G.show()                          # long time                             # needs sage.plot
      sage: G.is_planar()
      True
      sage: G.order()
      5
      sage: G.size()
      6

  Python

      >>> from sage.all import *
      >>> G = graphs.ButterflyGraph(); G
      Butterfly graph: Graph on 5 vertices
      >>> G.show()                          # long time                             # needs sage.plot
      >>> G.is_planar()
      True
      >>> G.order()
      5
      >>> G.size()
      6

  It has diameter 2, girth 3, and radius 1:

  Sage

      sage: G.diameter()
      2
      sage: G.girth()
      3
      sage: G.radius()
      1

  [Python]

      >>> from sage.all import *
      >>> G.diameter()
      2
      >>> G.girth()
      3
      >>> G.radius()
      1

  The butterfly graph is Eulerian, with chromatic number 3:

  Sage

      sage: G.is_eulerian()
      True
      sage: G.chromatic_number()
      3

  Python

      >>> from sage.all import *
      >>> G.is_eulerian()
      True
      >>> G.chromatic_number()
      3

sage.graphs.generators.basic.CircularLadderGraph(n, immutable=False)

  Return a circular ladder graph with \(2 * n\) nodes.

  A Circular ladder graph is a ladder graph that is connected at the ends, i.e.: a ladder bent around so that top meets bottom. Thus it can be described as two parallel cycle graphs connected at each corresponding node pair.

  PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the circular ladder graph is displayed as an inner and outer cycle pair, with the first \(n\) nodes drawn on the inner circle. The first (0) node is drawn at the top of the inner-circle, moving clockwise after that. The outer circle is drawn with the \((n+1)\)-th node at the top, then counterclockwise as well. When \(n == 2\), we rotate the outer circle by an angle of \(\pi/8\) to ensure that all edges are visible (otherwise the 4 vertices of the graph would be placed on a single line).

  INPUT:

  - `n` – nonnegative integer

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  EXAMPLES:

  Construct and show a circular ladder graph with 26 nodes:

  Sage

      sage: g = graphs.CircularLadderGraph(13)
      sage: g.show()                          # long time                             # needs sage.plot

  Python

      >>> from sage.all import *
      >>> g = graphs.CircularLadderGraph(Integer(13))
      >>> g.show()                          # long time                             # needs sage.plot

  Create several circular ladder graphs in a Sage graphics array:

  Sage

      sage: g = []
      sage: j = []
      sage: for i in range(9):
      ....:    k = graphs.CircularLadderGraph(i+3)
      ....:    g.append(k)
      sage: for i in range(3):                                                        # needs sage.plot
      ....:    n = []
      ....:    for m in range(3):
      ....:        n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
      ....:    j.append(n)
      sage: G = graphics_array(j)                                                     # needs sage.plot
      sage: G.show()                          # long time                             # needs sage.plot

  [Python]

      >>> from sage.all import *
      >>> g = []
      >>> j = []
      >>> for i in range(Integer(9)):
      ...    k = graphs.CircularLadderGraph(i+Integer(3))
      ...    g.append(k)
      >>> for i in range(Integer(3)):                                                        # needs sage.plot
      ...    n = []
      ...    for m in range(Integer(3)):
      ...        n.append(g[Integer(3)*i + m].plot(vertex_size=Integer(50), vertex_labels=False))
      ...    j.append(n)
      >>> G = graphics_array(j)                                                     # needs sage.plot
      >>> G.show()                          # long time                             # needs sage.plot

sage.graphs.generators.basic.ClawGraph(immutable=False)

  Return a claw graph.

  A claw graph is named for its shape. It is actually a complete bipartite graph with `(n1, n2) = (1, 3)`.

  PLOTTING: See `CompleteBipartiteGraph()`.

  INPUT:

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  EXAMPLES:

  Show a Claw graph:

  Sage

      sage: (graphs.ClawGraph()).show()       # long time                             # needs sage.plot

  Python

      >>> from sage.all import *
      >>> (graphs.ClawGraph()).show()       # long time                             # needs sage.plot

  Inspect a Claw graph:

  Sage

      sage: G = graphs.ClawGraph()
      sage: G
      Claw graph: Graph on 4 vertices

  [Python]

      >>> from sage.all import *
      >>> G = graphs.ClawGraph()
      >>> G
      Claw graph: Graph on 4 vertices

sage.graphs.generators.basic.CompleteBipartiteGraph(p, q, set_position=True, immutable=False, name=None)

  Return a Complete Bipartite Graph on \(p + q\) vertices.

  A Complete Bipartite Graph is a graph with its vertices partitioned into two groups, \(V_1 = \{0,...,p-1\}\) and \(V_2 = \{p,...,p+q-1\}\). Each \(u \in V_1\) is connected to every \(v \in V_2\).

  INPUT:

  - `p`, `q` – number of vertices in each side

  - `set_position` – boolean (default: `True`); if set to `True`, we assign positions to the vertices so that the set of cardinality \(p\) is on the line \(y=1\) and the set of cardinality \(q\) is on the line \(y=0\).

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  - `name` – string (default: `None`); used as the name of the returned graph when set

  PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, each complete bipartite graph will be displayed with the first \(p\) nodes on the top row (at \(y=1\)) from left to right. The remaining \(q\) nodes appear at \(y=0\), also from left to right. The shorter row (partition with fewer nodes) is stretched to the same length as the longer row, unless the shorter row has 1 node; in which case it is centered. The \(x\) values in the plot are in domain \([0, \max(p, q)]\).

  In the Complete Bipartite graph, there is a visual difference in using the spring-layout algorithm vs. the position dictionary used in this constructor. The position dictionary flattens the graph and separates the partitioned nodes, making it clear which nodes an edge is connected to. The Complete Bipartite graph plotted with the spring-layout algorithm tends to center the nodes in \(p\) (see `spring_med` in examples below), thus overlapping its nodes and edges, making it typically hard to decipher.

  Filling the position dictionary in advance adds \(O(n)\) to the constructor. Feel free to race the constructors below in the examples section. The much larger difference is the time added by the spring-layout algorithm when plotting. (Also shown in the example below). The spring model is typically described as \(O(n^3)\), as appears to be the case in the NetworkX source code.

  EXAMPLES:

  Two ways of constructing the complete bipartite graph, using different layout algorithms:

  Sage

      sage: # needs networkx
      sage: import networkx
      sage: n = networkx.complete_bipartite_graph(389, 157)   # long time
      sage: spring_big = Graph(n)             # long time
      sage: posdict_big = graphs.CompleteBipartiteGraph(389, 157)        # long time

  Python

      >>> from sage.all import *
      >>> # needs networkx
      >>> import networkx
      >>> n = networkx.complete_bipartite_graph(Integer(389), Integer(157))   # long time
      >>> spring_big = Graph(n)             # long time
      >>> posdict_big = graphs.CompleteBipartiteGraph(Integer(389), Integer(157))        # long time

  Compare the plotting:

  Sage

      sage: n = networkx.complete_bipartite_graph(11, 17)                             # needs networkx
      sage: spring_med = Graph(n)                                                     # needs networkx
      sage: posdict_med = graphs.CompleteBipartiteGraph(11, 17)

  [Python]

      >>> from sage.all import *
      >>> n = networkx.complete_bipartite_graph(Integer(11), Integer(17))                             # needs networkx
      >>> spring_med = Graph(n)                                                     # needs networkx
      >>> posdict_med = graphs.CompleteBipartiteGraph(Integer(11), Integer(17))

  Notice here how the spring-layout tends to center the nodes of \(n1\):

  Sage

      sage: spring_med.show()                 # long time                             # needs networkx
      sage: posdict_med.show()                # long time                             # needs sage.plot

  Python

      >>> from sage.all import *
      >>> spring_med.show()                 # long time                             # needs networkx
      >>> posdict_med.show()                # long time                             # needs sage.plot

  View many complete bipartite graphs with a Sage Graphics Array, with this constructor (i.e., the position dictionary filled):

  Sage

      sage: g = []
      sage: j = []
      sage: for i in range(9):
      ....:     k = graphs.CompleteBipartiteGraph(i+1,4)
      ....:     g.append(k)
      sage: for i in range(3):                                                        # needs sage.plot
      ....:     n = []
      ....:     for m in range(3):
      ....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
      ....:     j.append(n)
      sage: G = graphics_array(j)                                                     # needs sage.plot
      sage: G.show()                          # long time                             # needs sage.plot

  [Python]

      >>> from sage.all import *
      >>> g = []
      >>> j = []
      >>> for i in range(Integer(9)):
      ...     k = graphs.CompleteBipartiteGraph(i+Integer(1),Integer(4))
      ...     g.append(k)
      >>> for i in range(Integer(3)):                                                        # needs sage.plot
      ...     n = []
      ...     for m in range(Integer(3)):
      ...         n.append(g[Integer(3)*i + m].plot(vertex_size=Integer(50), vertex_labels=False))
      ...     j.append(n)
      >>> G = graphics_array(j)                                                     # needs sage.plot
      >>> G.show()                          # long time                             # needs sage.plot

  We compare to plotting with the spring-layout algorithm:

  Sage

      sage: # needs networkx sage.plot
      sage: g = []
      sage: j = []
      sage: for i in range(9):
      ....:     spr = networkx.complete_bipartite_graph(i+1,4)
      ....:     k = Graph(spr)
      ....:     g.append(k)
      sage: for i in range(3):
      ....:     n = []
      ....:     for m in range(3):
      ....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
      ....:     j.append(n)
      sage: G = graphics_array(j)
      sage: G.show()                          # long time

  Python

      >>> from sage.all import *
      >>> # needs networkx sage.plot
      >>> g = []
      >>> j = []
      >>> for i in range(Integer(9)):
      ...     spr = networkx.complete_bipartite_graph(i+Integer(1),Integer(4))
      ...     k = Graph(spr)
      ...     g.append(k)
      >>> for i in range(Integer(3)):
      ...     n = []
      ...     for m in range(Integer(3)):
      ...         n.append(g[Integer(3)*i + m].plot(vertex_size=Integer(50), vertex_labels=False))
      ...     j.append(n)
      >>> G = graphics_array(j)
      >>> G.show()                          # long time

  Issue #12155:

  Sage

      sage: graphs.CompleteBipartiteGraph(5,6).complement()
      complement(Complete bipartite graph of order 5+6): Graph on 11 vertices

  [Python]

      >>> from sage.all import *
      >>> graphs.CompleteBipartiteGraph(Integer(5),Integer(6)).complement()
      complement(Complete bipartite graph of order 5+6): Graph on 11 vertices

sage.graphs.generators.basic.CompleteGraph(n, immutable=False)

  Return a complete graph on \(n\) nodes.

  A Complete Graph is a graph in which all nodes are connected to all other nodes.

  INPUT:

  - `n` – nonnegative integer; the number of vertices of the complete graph

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, each complete graph will be displayed with the first (0) node at the top, with the rest following in a counterclockwise manner.

  In the complete graph, there is a big difference visually in using the spring-layout algorithm vs. the position dictionary used in this constructor. The position dictionary flattens the graph, making it clear which nodes an edge is connected to. But the complete graph offers a good example of how the spring-layout works. The edges push outward (everything is connected), causing the graph to appear as a 3-dimensional pointy ball. (See examples below).

  EXAMPLES:

  We view many Complete graphs with a Sage Graphics Array, first with this constructor (i.e., the position dictionary filled):

  Sage

      sage: g = []
      sage: j = []
      sage: for i in range(9):
      ....:     k = graphs.CompleteGraph(i+3)
      ....:     g.append(k)
      sage: for i in range(3):
      ....:     n = []
      ....:     for m in range(3):
      ....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
      ....:     j.append(n)
      sage: G = graphics_array(j)
      sage: G.show()                          # long time

  Python

      >>> from sage.all import *
      >>> g = []
      >>> j = []
      >>> for i in range(Integer(9)):
      ...     k = graphs.CompleteGraph(i+Integer(3))
      ...     g.append(k)
      >>> for i in range(Integer(3)):
      ...     n = []
      ...     for m in range(Integer(3)):
      ...         n.append(g[Integer(3)*i + m].plot(vertex_size=Integer(50), vertex_labels=False))
      ...     j.append(n)
      >>> G = graphics_array(j)
      >>> G.show()                          # long time

  We compare to plotting with the spring-layout algorithm:

  Sage

      sage: # needs networkx sage.plot
      sage: import networkx
      sage: g = []
      sage: j = []
      sage: for i in range(9):
      ....:     spr = networkx.complete_graph(i+3)
      ....:     k = Graph(spr)
      ....:     g.append(k)
      sage: for i in range(3):
      ....:     n = []
      ....:     for m in range(3):
      ....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
      ....:     j.append(n)
      sage: G = graphics_array(j)
      sage: G.show()                          # long time

  [Python]

      >>> from sage.all import *
      >>> # needs networkx sage.plot
      >>> import networkx
      >>> g = []
      >>> j = []
      >>> for i in range(Integer(9)):
      ...     spr = networkx.complete_graph(i+Integer(3))
      ...     k = Graph(spr)
      ...     g.append(k)
      >>> for i in range(Integer(3)):
      ...     n = []
      ...     for m in range(Integer(3)):
      ...         n.append(g[Integer(3)*i + m].plot(vertex_size=Integer(50), vertex_labels=False))
      ...     j.append(n)
      >>> G = graphics_array(j)
      >>> G.show()                          # long time

  Compare the constructors (results will vary):

  Sage

      sage: # needs networkx
      sage: import networkx
      sage: t = cputime()
      sage: n = networkx.complete_graph(389); spring389 = Graph(n)
      sage: cputime(t)  # random
      0.59203700000000126
      sage: t = cputime()
      sage: posdict389 = graphs.CompleteGraph(389)
      sage: cputime(t)  # random
      0.6680419999999998

  Python

      >>> from sage.all import *
      >>> # needs networkx
      >>> import networkx
      >>> t = cputime()
      >>> n = networkx.complete_graph(Integer(389)); spring389 = Graph(n)
      >>> cputime(t)  # random
      0.59203700000000126
      >>> t = cputime()
      >>> posdict389 = graphs.CompleteGraph(Integer(389))
      >>> cputime(t)  # random
      0.6680419999999998

  We compare plotting:

  Sage

      sage: # needs networkx
      sage: import networkx
      sage: n = networkx.complete_graph(23)
      sage: spring23 = Graph(n)
      sage: posdict23 = graphs.CompleteGraph(23)
      sage: spring23.show()                   # long time                             # needs sage.plot
      sage: posdict23.show()                  # long time                             # needs sage.plot

  [Python]

      >>> from sage.all import *
      >>> # needs networkx
      >>> import networkx
      >>> n = networkx.complete_graph(Integer(23))
      >>> spring23 = Graph(n)
      >>> posdict23 = graphs.CompleteGraph(Integer(23))
      >>> spring23.show()                   # long time                             # needs sage.plot
      >>> posdict23.show()                  # long time                             # needs sage.plot

sage.graphs.generators.basic.CompleteMultipartiteGraph(L, immutable=False)

  Return a complete multipartite graph.

  INPUT:

  - `L` – list of integers; the respective sizes of the components

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  PLOTTING: Produce a layout of the vertices so that vertices in the same vertex set are adjacent and clearly separated from vertices in other vertex sets.

  This is done by calculating the vertices of an \(r\)-gon then calculating the slope between adjacent vertices. We then ‘walk’ around the \(r\)-gon placing graph vertices in regular intervals between adjacent vertices of the \(r\)-gon.

  Makes a nicely organized graph like in this picture: https://commons.wikimedia.org/wiki/File:Turan_13-4.svg

  EXAMPLES:

  A complete tripartite graph with sets of sizes \(5, 6, 8\):

  Sage

      sage: g = graphs.CompleteMultipartiteGraph([5, 6, 8]); g
      Multipartite Graph with set sizes [5, 6, 8]: Graph on 19 vertices

  Python

      >>> from sage.all import *
      >>> g = graphs.CompleteMultipartiteGraph([Integer(5), Integer(6), Integer(8)]); g
      Multipartite Graph with set sizes [5, 6, 8]: Graph on 19 vertices

  It clearly has a chromatic number of 3:

  Sage

      sage: g.chromatic_number()
      3

  [Python]

      >>> from sage.all import *
      >>> g.chromatic_number()
      3

sage.graphs.generators.basic.CorrelationGraph(seqs, alpha, include_anticorrelation, immutable=False)

  Return a correlation graph with a node per sequence in `seqs`.

  Edges are added between nodes where the corresponding sequences have a correlation coefficient greater than alpha.

  If `include_anticorrelation` is `True`, then edges are also added between nodes with correlation coefficient less than `-alpha`.

  INPUT:

  - `seqs` – list of sequences, that is a list of lists

  - `alpha` – float; threshold on the correlation coefficient between two sequences for adding an edge

  - `include_anticorrelation` – boolean; whether to add edges between nodes with correlation coefficient less than `-alpha` or not

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  EXAMPLES:

  Sage

      sage: # needs numpy
      sage: from sage.graphs.generators.basic import CorrelationGraph
      sage: data = [[1,2,3], [4,5,6], [7,8,9999]]
      sage: CG1 = CorrelationGraph(data, 0.9, False)
      sage: CG2 = CorrelationGraph(data, 0.9, True)
      sage: CG3 = CorrelationGraph(data, 0.1, True)
      sage: CG1.edges(sort=False)
      [(0, 0, None), (0, 1, None), (1, 1, None), (2, 2, None)]
      sage: CG2.edges(sort=False)
      [(0, 0, None), (0, 1, None), (1, 1, None), (2, 2, None)]
      sage: CG3.edges(sort=False)
      [(0, 0, None), (0, 1, None), (0, 2, None), (1, 1, None), (1, 2, None), (2, 2, None)]

  Python

      >>> from sage.all import *
      >>> # needs numpy
      >>> from sage.graphs.generators.basic import CorrelationGraph
      >>> data = [[Integer(1),Integer(2),Integer(3)], [Integer(4),Integer(5),Integer(6)], [Integer(7),Integer(8),Integer(9999)]]
      >>> CG1 = CorrelationGraph(data, RealNumber('0.9'), False)
      >>> CG2 = CorrelationGraph(data, RealNumber('0.9'), True)
      >>> CG3 = CorrelationGraph(data, RealNumber('0.1'), True)
      >>> CG1.edges(sort=False)
      [(0, 0, None), (0, 1, None), (1, 1, None), (2, 2, None)]
      >>> CG2.edges(sort=False)
      [(0, 0, None), (0, 1, None), (1, 1, None), (2, 2, None)]
      >>> CG3.edges(sort=False)
      [(0, 0, None), (0, 1, None), (0, 2, None), (1, 1, None), (1, 2, None), (2, 2, None)]

sage.graphs.generators.basic.CycleGraph(n, immutable=False)

  Return a cycle graph with \(n\) nodes.

  A cycle graph is a basic structure which is also typically called an \(n\)-gon.

  INPUT:

  - `n` – nonnegative integer; the number of vertices of the cycle graph

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, each cycle graph will be displayed with the first (0) node at the top, with the rest following in a counterclockwise manner.

  The cycle graph is a good opportunity to compare efficiency of filling a position dictionary vs. using the spring-layout algorithm for plotting. Because the cycle graph is very symmetric, the resulting plots should be similar (in cases of small \(n\)).

  Filling the position dictionary in advance adds \(O(n)\) to the constructor.

  EXAMPLES:

  Compare plotting using the predefined layout and networkx:

  Sage

      sage: # needs networkx sage.plot
      sage: import networkx
      sage: n = networkx.cycle_graph(23)
      sage: spring23 = Graph(n)
      sage: posdict23 = graphs.CycleGraph(23)
      sage: spring23.show()                   # long time
      sage: posdict23.show()                  # long time

  Python

      >>> from sage.all import *
      >>> # needs networkx sage.plot
      >>> import networkx
      >>> n = networkx.cycle_graph(Integer(23))
      >>> spring23 = Graph(n)
      >>> posdict23 = graphs.CycleGraph(Integer(23))
      >>> spring23.show()                   # long time
      >>> posdict23.show()                  # long time

  We next view many cycle graphs as a Sage graphics array. First we use the `CycleGraph` constructor, which fills in the position dictionary:

  Sage

      sage: # needs networkx sage.plot
      sage: g = []
      sage: j = []
      sage: for i in range(9):
      ....:     k = graphs.CycleGraph(i+3)
      ....:     g.append(k)
      sage: for i in range(3):
      ....:     n = []
      ....:     for m in range(3):
      ....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
      ....:     j.append(n)
      sage: G = graphics_array(j)
      sage: G.show()                          # long time

  [Python]

      >>> from sage.all import *
      >>> # needs networkx sage.plot
      >>> g = []
      >>> j = []
      >>> for i in range(Integer(9)):
      ...     k = graphs.CycleGraph(i+Integer(3))
      ...     g.append(k)
      >>> for i in range(Integer(3)):
      ...     n = []
      ...     for m in range(Integer(3)):
      ...         n.append(g[Integer(3)*i + m].plot(vertex_size=Integer(50), vertex_labels=False))
      ...     j.append(n)
      >>> G = graphics_array(j)
      >>> G.show()                          # long time

  Compare to plotting with the spring-layout algorithm:

  Sage

      sage: # needs networkx sage.plot
      sage: g = []
      sage: j = []
      sage: for i in range(9):
      ....:     spr = networkx.cycle_graph(i+3)
      ....:     k = Graph(spr)
      ....:     g.append(k)
      sage: for i in range(3):
      ....:     n = []
      ....:     for m in range(3):
      ....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
      ....:     j.append(n)
      sage: G = graphics_array(j)
      sage: G.show()                          # long time

  Python

      >>> from sage.all import *
      >>> # needs networkx sage.plot
      >>> g = []
      >>> j = []
      >>> for i in range(Integer(9)):
      ...     spr = networkx.cycle_graph(i+Integer(3))
      ...     k = Graph(spr)
      ...     g.append(k)
      >>> for i in range(Integer(3)):
      ...     n = []
      ...     for m in range(Integer(3)):
      ...         n.append(g[Integer(3)*i + m].plot(vertex_size=Integer(50), vertex_labels=False))
      ...     j.append(n)
      >>> G = graphics_array(j)
      >>> G.show()                          # long time

sage.graphs.generators.basic.DartGraph(immutable=False)

  Return a dart graph with 5 nodes.

  INPUT:

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the dart graph is drawn as a dart, with the sharp part on the bottom.

  EXAMPLES:

  Construct and show a dart graph:

  Sage

      sage: g = graphs.DartGraph()
      sage: g.show()                          # long time                             # needs sage.plot

  Python

      >>> from sage.all import *
      >>> g = graphs.DartGraph()
      >>> g.show()                          # long time                             # needs sage.plot

sage.graphs.generators.basic.DiamondGraph(immutable=False)

  Return a diamond graph with 4 nodes.

  A diamond graph is a square with one pair of diagonal nodes connected.

  INPUT:

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the diamond graph is drawn as a diamond, with the first node on top, second on the left, third on the right, and fourth on the bottom; with the second and third node connected.

  EXAMPLES:

  Construct and show a diamond graph:

  Sage

      sage: g = graphs.DiamondGraph()
      sage: g.show()                          # long time                             # needs sage.plot

  Python

      >>> from sage.all import *
      >>> g = graphs.DiamondGraph()
      >>> g.show()                          # long time                             # needs sage.plot

sage.graphs.generators.basic.EmptyGraph(immutable=False)

  Return an empty graph (0 nodes and 0 edges).

  This is useful for constructing graphs by adding edges and vertices individually or in a loop.

  INPUT:

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  PLOTTING: When plotting, this graph will use the default spring-layout algorithm, unless a position dictionary is specified.

  EXAMPLES:

  Add one vertex to an empty graph and then show:

  Sage

      sage: empty1 = graphs.EmptyGraph()
      sage: empty1.add_vertex()
      0
      sage: empty1.show()                     # long time                             # needs sage.plot

  Python

      >>> from sage.all import *
      >>> empty1 = graphs.EmptyGraph()
      >>> empty1.add_vertex()
      0
      >>> empty1.show()                     # long time                             # needs sage.plot

  Use for loops to build a graph from an empty graph:

  Sage

      sage: empty2 = graphs.EmptyGraph()
      sage: for i in range(5):
      ....:     empty2.add_vertex()  # add 5 nodes, labeled 0-4
      0
      1
      2
      3
      4
      sage: for i in range(3):
      ....:     empty2.add_edge(i,i+1)  # add edges {[0:1],[1:2],[2:3]}
      sage: for i in range(1, 4):
      ....:     empty2.add_edge(4,i)  # add edges {[1:4],[2:4],[3:4]}
      sage: empty2.show()                     # long time                             # needs sage.plot

  [Python]

      >>> from sage.all import *
      >>> empty2 = graphs.EmptyGraph()
      >>> for i in range(Integer(5)):
      ...     empty2.add_vertex()  # add 5 nodes, labeled 0-4
      0
      1
      2
      3
      4
      >>> for i in range(Integer(3)):
      ...     empty2.add_edge(i,i+Integer(1))  # add edges {[0:1],[1:2],[2:3]}
      >>> for i in range(Integer(1), Integer(4)):
      ...     empty2.add_edge(Integer(4),i)  # add edges {[1:4],[2:4],[3:4]}
      >>> empty2.show()                     # long time                             # needs sage.plot

sage.graphs.generators.basic.ForkGraph(immutable=False)

  Return a fork graph with 5 nodes.

  A fork graph, sometimes also called chair graph, is 5 vertex tree.

  INPUT:

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the fork graph is drawn as a fork, with the sharp part on the bottom.

  EXAMPLES:

  Construct and show a fork graph:

  Sage

      sage: g = graphs.ForkGraph()
      sage: g.show()                          # long time                             # needs sage.plot

  Python

      >>> from sage.all import *
      >>> g = graphs.ForkGraph()
      >>> g.show()                          # long time                             # needs sage.plot

sage.graphs.generators.basic.GemGraph(immutable=False)

  Return a gem graph with 5 nodes.

  A gem graph is a fan graph (4,1).

  INPUT:

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the gem graph is drawn as a gem, with the sharp part on the bottom.

  EXAMPLES:

  Construct and show a gem graph:

  Sage

      sage: g = graphs.GemGraph()
      sage: g.show()                          # long time                             # needs sage.plot

  Python

      >>> from sage.all import *
      >>> g = graphs.GemGraph()
      >>> g.show()                          # long time                             # needs sage.plot

sage.graphs.generators.basic.Grid2dGraph(p, q, set_positions=True, immutable=False, name=None)

  Return a \(2\)-dimensional grid graph with \(p \times q\) nodes (\(p\) rows and \(q\) columns).

  A 2d grid graph resembles a \(2\) dimensional grid. All inner nodes are connected to their \(4\) neighbors. Outer (non-corner) nodes are connected to their \(3\) neighbors. Corner nodes are connected to their 2 neighbors.

  INPUT:

  - `p`, `q` – two positive integers

  - `set_positions` – boolean (default: `True`); whether to set the position of the nodes

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  - `name` – string (default: `None`); used as the name of the returned graph when set

  PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, nodes are labelled in (row, column) pairs with \((0, 0)\) in the top left corner. Edges will always be horizontal and vertical - another advantage of filling the position dictionary.

  EXAMPLES:

  Construct and show a grid 2d graph Rows = \(5\), Columns = \(7\):

  Sage

      sage: g = graphs.Grid2dGraph(5,7)
      sage: g.show()                          # long time                             # needs sage.plot

  Python

      >>> from sage.all import *
      >>> g = graphs.Grid2dGraph(Integer(5),Integer(7))
      >>> g.show()                          # long time                             # needs sage.plot

sage.graphs.generators.basic.GridGraph(dim_list, immutable=False)

  Return an \(n\)-dimensional grid graph.

  INPUT:

  - `dim_list` – list of integers representing the number of nodes to extend in each dimension

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  PLOTTING: When plotting, this graph will use the default spring-layout algorithm, unless a position dictionary is specified.

  EXAMPLES:

  Sage

      sage: G = graphs.GridGraph([2,3,4])
      sage: G.show()                          # long time                             # needs sage.plot

  Python

      >>> from sage.all import *
      >>> G = graphs.GridGraph([Integer(2),Integer(3),Integer(4)])
      >>> G.show()                          # long time                             # needs sage.plot

  Sage

      sage: C = graphs.CubeGraph(4)
      sage: G = graphs.GridGraph([2,2,2,2])
      sage: C.show()                          # long time                             # needs sage.plot
      sage: G.show()                          # long time                             # needs sage.plot

  [Python]

      >>> from sage.all import *
      >>> C = graphs.CubeGraph(Integer(4))
      >>> G = graphs.GridGraph([Integer(2),Integer(2),Integer(2),Integer(2)])
      >>> C.show()                          # long time                             # needs sage.plot
      >>> G.show()                          # long time                             # needs sage.plot

sage.graphs.generators.basic.HouseGraph(immutable=False)

  Return a house graph with 5 nodes.

  A house graph is named for its shape. It is a triangle (roof) over a square (walls).

  PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the house graph is drawn with the first node in the lower-left corner of the house, the second in the lower-right corner of the house. The third node is in the upper-left corner connecting the roof to the wall, and the fourth is in the upper-right corner connecting the roof to the wall. The fifth node is the top of the roof, connected only to the third and fourth.

  INPUT:

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  EXAMPLES:

  Construct and show a house graph:

  Sage

      sage: g = graphs.HouseGraph()
      sage: g.show()                          # long time                             # needs sage.plot

  Python

      >>> from sage.all import *
      >>> g = graphs.HouseGraph()
      >>> g.show()                          # long time                             # needs sage.plot

sage.graphs.generators.basic.HouseXGraph(immutable=False)

  Return a house X graph with 5 nodes.

  A house X graph is a house graph with two additional edges. The upper-right corner is connected to the lower-left. And the upper-left corner is connected to the lower-right.

  PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the house X graph is drawn with the first node in the lower-left corner of the house, the second in the lower-right corner of the house. The third node is in the upper-left corner connecting the roof to the wall, and the fourth is in the upper-right corner connecting the roof to the wall. The fifth node is the top of the roof, connected only to the third and fourth.

  INPUT:

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  EXAMPLES:

  Construct and show a house X graph:

  Sage

      sage: g = graphs.HouseXGraph()
      sage: g.show()                          # long time                             # needs sage.plot

  Python

      >>> from sage.all import *
      >>> g = graphs.HouseXGraph()
      >>> g.show()                          # long time                             # needs sage.plot

sage.graphs.generators.basic.LadderGraph(n, immutable=False)

  Return a ladder graph with \(2 * n\) nodes.

  A ladder graph is a basic structure that is typically displayed as a ladder, i.e.: two parallel path graphs connected at each corresponding node pair.

  PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, each ladder graph will be displayed horizontally, with the first n nodes displayed left to right on the top horizontal line.

  INPUT:

  - `n` – a nonnegative integer; number of nodes is \(2n\)

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  EXAMPLES:

  Construct and show a ladder graph with 14 nodes:

  Sage

      sage: g = graphs.LadderGraph(7)
      sage: g.show()                          # long time                             # needs sage.plot

  Python

      >>> from sage.all import *
      >>> g = graphs.LadderGraph(Integer(7))
      >>> g.show()                          # long time                             # needs sage.plot

  Create several ladder graphs in a Sage graphics array:

  Sage

      sage: g = []
      sage: j = []
      sage: for i in range(9):
      ....:     k = graphs.LadderGraph(i+2)
      ....:     g.append(k)
      sage: for i in range(3):
      ....:     n = []
      ....:     for m in range(3):
      ....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
      ....:     j.append(n)
      sage: G = graphics_array(j)
      sage: G.show()                          # long time

  [Python]

      >>> from sage.all import *
      >>> g = []
      >>> j = []
      >>> for i in range(Integer(9)):
      ...     k = graphs.LadderGraph(i+Integer(2))
      ...     g.append(k)
      >>> for i in range(Integer(3)):
      ...     n = []
      ...     for m in range(Integer(3)):
      ...         n.append(g[Integer(3)*i + m].plot(vertex_size=Integer(50), vertex_labels=False))
      ...     j.append(n)
      >>> G = graphics_array(j)
      >>> G.show()                          # long time

sage.graphs.generators.basic.MoebiusLadderGraph(n, immutable=False)

  Return a Möbius ladder graph with \(2n\) nodes

  A Möbius ladder graph of order \(2n\) is a ladder graph of the same order that is connected at the ends with a single twist, i.e., a ladder graph bent around so that top meets bottom with a single twist. Alternatively, it can be described as a single cycle graph (of order \(2n\)) with the addition of edges (called \(rungs\)) joining the antipodal pairs of nodes. Also, note that the Möbius ladder graph `graphs.MoebiusLadderGraph(n)` is precisely the same graph as the circulant graph `graphs.CirculantGraph(2 * n, [1, n])`.

  PLOTTING:

  Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, each Möbius ladder graph will be displayed with the first (0) node at the top, with the rest following in a counterclockwise manner.

  INPUT:

  - `n` – a nonnegative integer; number of nodes is \(2n\)

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  OUTPUT:

  - `G` – a Möbius ladder graph of order \(2n\); note that a `ValueError` is returned if \(n < 0\)

  EXAMPLES:

  Construct and show a Möbius ladder graph with 26 nodes:

  Sage

      sage: g = graphs.MoebiusLadderGraph(13)
      sage: g.show()                          # long time                             # needs sage.plot

  Python

      >>> from sage.all import *
      >>> g = graphs.MoebiusLadderGraph(Integer(13))
      >>> g.show()                          # long time                             # needs sage.plot

  Create several Möbius ladder graphs in a Sage graphics array:

  Sage

      sage: g = []
      sage: j = []
      sage: for i in range(9):
      ....:    k = graphs.MoebiusLadderGraph(i+3)
      ....:    g.append(k)
      sage: for i in range(3):
      ....:    n = []
      ....:    for m in range(3):
      ....:        n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
      ....:    j.append(n)
      sage: G = graphics_array(j)
      sage: G.show()                          # long time

  [Python]

      >>> from sage.all import *
      >>> g = []
      >>> j = []
      >>> for i in range(Integer(9)):
      ...    k = graphs.MoebiusLadderGraph(i+Integer(3))
      ...    g.append(k)
      >>> for i in range(Integer(3)):
      ...    n = []
      ...    for m in range(Integer(3)):
      ...        n.append(g[Integer(3)*i + m].plot(vertex_size=Integer(50), vertex_labels=False))
      ...    j.append(n)
      >>> G = graphics_array(j)
      >>> G.show()                          # long time

  REFERENCES:

  - Wikipedia article Möbius_ladder

  See also

  `LadderGraph()`, `CircularLadderGraph()`, `CirculantGraph()`

  AUTHORS:

  - Janmenjaya Panda (2024-05-26)

sage.graphs.generators.basic.PathGraph(n, pos=None, immutable=False, name=None)

  Return a path graph with \(n\) nodes.

  A path graph is a graph where all inner nodes are connected to their two neighbors and the two end-nodes are connected to their one inner neighbors (i.e.: a cycle graph without the first and last node connected).

  INPUT:

  - `n` – nonnegative integer; number of nodes of the path graph

  - `pos` – string (default: `None`); indicates the embedding to use between ‘circle’, ‘line’ or the default algorithm. See the plotting section below for more detail.

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  - `name` – string (default: `None`); used as the name of the returned graph when set

  PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the graph may be drawn in one of two ways: The ‘line’ argument will draw the graph in a horizontal line (left to right) if there are less than 11 nodes. Otherwise the ‘line’ argument will append horizontal lines of length 10 nodes below, alternating left to right and right to left. The ‘circle’ argument will cause the graph to be drawn in a cycle-shape, with the first node at the top and then about the circle in a clockwise manner. By default (without an appropriate string argument) the graph will be drawn as a ‘circle’ if \(10 < n < 41\) and as a ‘line’ for all other \(n\).

  EXAMPLES:

  Show default drawing by size: ‘line’: \(n \leq 10\):

  Sage

      sage: p = graphs.PathGraph(10)
      sage: p.show()                          # long time                             # needs sage.plot

  Python

      >>> from sage.all import *
      >>> p = graphs.PathGraph(Integer(10))
      >>> p.show()                          # long time                             # needs sage.plot

  ‘circle’: \(10 < n < 41\):

  Sage

      sage: q = graphs.PathGraph(25)
      sage: q.show()                          # long time                             # needs sage.plot

  [Python]

      >>> from sage.all import *
      >>> q = graphs.PathGraph(Integer(25))
      >>> q.show()                          # long time                             # needs sage.plot

  ‘line’: \(n \geq 41\):

  Sage

      sage: r = graphs.PathGraph(55)
      sage: r.show()                          # long time                             # needs sage.plot

  Python

      >>> from sage.all import *
      >>> r = graphs.PathGraph(Integer(55))
      >>> r.show()                          # long time                             # needs sage.plot

  Override the default drawing:

  Sage

      sage: s = graphs.PathGraph(5,'circle')
      sage: s.show()                          # long time                             # needs sage.plot

  [Python]

      >>> from sage.all import *
      >>> s = graphs.PathGraph(Integer(5),'circle')
      >>> s.show()                          # long time                             # needs sage.plot

sage.graphs.generators.basic.StarGraph(n, immutable=False)

  Return a star graph with \(n + 1\) nodes.

  A Star graph is a basic structure where one node is connected to all other nodes.

  INPUT:

  - `n` – a nonnegative integer; number of nodes is \(n + 1\)

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, each star graph will be displayed with the first (0) node in the center, the second node (1) at the top, with the rest following in a counterclockwise manner. (0) is the node connected to all other nodes.

  The star graph is a good opportunity to compare efficiency of filling a position dictionary vs. using the spring-layout algorithm for plotting. As far as display, the spring-layout should push all other nodes away from the (0) node, and thus look very similar to this constructor’s positioning.

  EXAMPLES:

  Sage

      sage: import networkx                                                           # needs networkx

  Python

      >>> from sage.all import *
      >>> import networkx                                                           # needs networkx

  Compare the plots:

  Sage

      sage: # needs networkx sage.plot
      sage: n = networkx.star_graph(23)
      sage: spring23 = Graph(n)
      sage: posdict23 = graphs.StarGraph(23)
      sage: spring23.show()                   # long time
      sage: posdict23.show()  # long time

  [Python]

      >>> from sage.all import *
      >>> # needs networkx sage.plot
      >>> n = networkx.star_graph(Integer(23))
      >>> spring23 = Graph(n)
      >>> posdict23 = graphs.StarGraph(Integer(23))
      >>> spring23.show()                   # long time
      >>> posdict23.show()  # long time

  View many star graphs as a Sage Graphics Array

  With this constructor (i.e., the position dictionary filled)

  Sage

      sage: g = []
      sage: j = []
      sage: for i in range(9):
      ....:     k = graphs.StarGraph(i+3)
      ....:     g.append(k)
      sage: for i in range(3):
      ....:     n = []
      ....:     for m in range(3):
      ....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
      ....:     j.append(n)
      sage: G = graphics_array(j)
      sage: G.show()                          # long time

  Python

      >>> from sage.all import *
      >>> g = []
      >>> j = []
      >>> for i in range(Integer(9)):
      ...     k = graphs.StarGraph(i+Integer(3))
      ...     g.append(k)
      >>> for i in range(Integer(3)):
      ...     n = []
      ...     for m in range(Integer(3)):
      ...         n.append(g[Integer(3)*i + m].plot(vertex_size=Integer(50), vertex_labels=False))
      ...     j.append(n)
      >>> G = graphics_array(j)
      >>> G.show()                          # long time

  Compared to plotting with the spring-layout algorithm

  Sage

      sage: # needs networkx sage.plot
      sage: g = []
      sage: j = []
      sage: for i in range(9):
      ....:     spr = networkx.star_graph(i+3)
      ....:     k = Graph(spr)
      ....:     g.append(k)
      sage: for i in range(3):
      ....:     n = []
      ....:     for m in range(3):
      ....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
      ....:     j.append(n)
      sage: G = graphics_array(j)
      sage: G.show()                          # long time

  [Python]

      >>> from sage.all import *
      >>> # needs networkx sage.plot
      >>> g = []
      >>> j = []
      >>> for i in range(Integer(9)):
      ...     spr = networkx.star_graph(i+Integer(3))
      ...     k = Graph(spr)
      ...     g.append(k)
      >>> for i in range(Integer(3)):
      ...     n = []
      ...     for m in range(Integer(3)):
      ...         n.append(g[Integer(3)*i + m].plot(vertex_size=Integer(50), vertex_labels=False))
      ...     j.append(n)
      >>> G = graphics_array(j)
      >>> G.show()                          # long time

sage.graphs.generators.basic.Toroidal6RegularGrid2dGraph(p, q, immutable=False)

  Return a toroidal 6-regular grid.

  The toroidal 6-regular grid is a 6-regular graph on \(p \times q\) vertices and its elements have coordinates \((i, j)\) for \(i \in \{0...p-1\}\) and \(j \in \{0...q-1\}\).

  Its edges are those of the `ToroidalGrid2dGraph()`, to which are added the edges between \((i, j)\) and \(((i + 1) \% p, (j + 1) \% q)\).

  INPUT:

  - `p`, `q` – integers

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  EXAMPLES:

  The toroidal 6-regular grid on \(25\) elements:

  Sage

      sage: g = graphs.Toroidal6RegularGrid2dGraph(5,5)
      sage: g.is_regular(k=6)
      True
      sage: g.is_vertex_transitive()                                                  # needs sage.groups
      True
      sage: g.line_graph().is_vertex_transitive()                                     # needs sage.groups
      True
      sage: g.automorphism_group().cardinality()                                      # needs sage.groups
      300
      sage: g.is_hamiltonian()                                                        # needs sage.numerical.mip
      True

  Python

      >>> from sage.all import *
      >>> g = graphs.Toroidal6RegularGrid2dGraph(Integer(5),Integer(5))
      >>> g.is_regular(k=Integer(6))
      True
      >>> g.is_vertex_transitive()                                                  # needs sage.groups
      True
      >>> g.line_graph().is_vertex_transitive()                                     # needs sage.groups
      True
      >>> g.automorphism_group().cardinality()                                      # needs sage.groups
      300
      >>> g.is_hamiltonian()                                                        # needs sage.numerical.mip
      True

sage.graphs.generators.basic.ToroidalGrid2dGraph(p, q, immutable=False)

  Return a toroidal 2-dimensional grid graph with \(p \times q\) nodes (\(p\) rows and \(q\) columns).

  The toroidal 2-dimensional grid with parameters \(p,q\) is the 2-dimensional grid graph with identical parameters to which are added the edges \(((i, 0), (i, q - 1))\) and \(((0, i), (p - 1, i))\).

  INPUT:

  - `p, q` – nonnegative integers; the sides of the toroidal grid

  - `immutable` – boolean (default: `False`); whether to return an immutable or a mutable graph

  EXAMPLES:

  The toroidal 2-dimensional grid is a regular graph, while the usual 2-dimensional grid is not

  Sage

      sage: tgrid = graphs.ToroidalGrid2dGraph(8,9)
      sage: print(tgrid)
      Toroidal 2D Grid Graph with parameters 8,9
      sage: grid = graphs.Grid2dGraph(8,9)
      sage: grid.is_regular()
      False
      sage: tgrid.is_regular()
      True

  Python

      >>> from sage.all import *
      >>> tgrid = graphs.ToroidalGrid2dGraph(Integer(8),Integer(9))
      >>> print(tgrid)
      Toroidal 2D Grid Graph with parameters 8,9
      >>> grid = graphs.Grid2dGraph(Integer(8),Integer(9))
      >>> grid.is_regular()
      False
      >>> tgrid.is_regular()
      True