def Fan(self, n, deg_three_verts=False):
"""
Sandpile on the Fan graph with a total of `n` vertices.
INPUT:
- ``n`` -- a non-negative integer
OUTPUT:
- Sandpile
EXAMPLES::
sage: f = sandpiles.Fan(10)
sage: f.group_order() == fibonacci(18)
True
sage: f = sandpiles.Fan(10,True) # all nonsink vertices have deg 3
sage: f.group_order() == fibonacci(20)
True
"""
f = graphs.WheelGraph(n)
if n > 2:
f.delete_edge(1, n - 1)
if deg_three_verts:
f.allow_multiple_edges(True)
f.add_edges([(0, 1), (0, n - 1)])
return Sandpile(f, 0)
elif n == 1:
return Sandpile(f, 0)
elif n == 2:
if deg_three_verts:
return Sandpile({0: {1: 3}, 1: {0: 3}})
else:
return Sandpile(f, 0)
def Cycle(self, n):
"""
Sandpile on the cycle graph with `n` vertices.
INPUT:
- ``n`` -- a non-negative integer
OUTPUT:
- Sandpile
EXAMPLES::
sage: s = sandpiles.Cycle(4)
sage: s.edges()
[(0, 1, 1),
(0, 3, 1),
(1, 0, 1),
(1, 2, 1),
(2, 1, 1),
(2, 3, 1),
(3, 0, 1),
(3, 2, 1)]
"""
return Sandpile(graphs.CycleGraph(n), 0)
def Grid(self, m, n):
"""
Sandpile on the diamond graph.
INPUT:
- ``m``, ``n`` -- negative integers
OUTPUT:
- Sandpile
EXAMPLES::
sage: s = sandpiles.Grid(2,3)
sage: s.vertices()
[(0, 0), (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)]
sage: s.invariant_factors()
[1, 1, 1, 1, 1, 2415]
sage: s = sandpiles.Grid(1,1)
sage: s.dict()
{(0, 0): {(1, 1): 4}, (1, 1): {(0, 0): 4}}
"""
G = graphs.Grid2dGraph(m + 2, n + 2)
G.allow_multiple_edges(
True) # to ensure each vertex ends up with degree 4
V = [(i, j) for i in [0, m + 1]
for j in range(n + 2)] + [(i, j) for j in [0, n + 1]
for i in range(m + 2)]
G.merge_vertices(V)
return Sandpile(G, (0, 0))
def Wheel(self, n):
"""
Sandpile on the wheel graph with a total of `n` vertices.
INPUT:
- ``n`` -- a non-negative integer
OUTPUT:
- Sandpile
EXAMPLES::
sage: w = sandpiles.Wheel(6)
sage: w.invariant_factors()
[1, 1, 1, 11, 11]
"""
return Sandpile(graphs.WheelGraph(n), 0)
def House(self):
"""
Sandpile on the House graph.
INPUT:
None
OUTPUT:
- Sandpile
EXAMPLES::
sage: s = sandpiles.House()
sage: s.invariant_factors()
[1, 1, 1, 11]
"""
return Sandpile(graphs.HouseGraph(), 0)
def Diamond(self):
"""
Sandpile on the diamond graph.
INPUT:
None
OUTPUT:
- Sandpile
EXAMPLES::
sage: s = sandpiles.Diamond()
sage: s.invariant_factors()
[1, 1, 8]
"""
return Sandpile(graphs.DiamondGraph(), 0)
def Complete(self, n):
"""
The complete sandpile graph with `n` vertices.
INPUT:
- ``n`` -- positive integer
OUTPUT:
- Sandpile
EXAMPLES::
sage: s = sandpiles.Complete(4)
sage: s.group_order()
16
sage: sandpiles.Complete(3) == sandpiles.Cycle(3)
True
"""
return Sandpile(graphs.CompleteGraph(n), 0)