Coverage for local/lib/python2.7/site-packages/sage/sandpiles/examples.py : 85%

# -*- coding: utf-8 -*- 

""" 

Examples of Sandpile 

AUTHORS: 

- David Perkinson (2015-05) [Using `examples.py` from homology as 

template.] 

This file constructs some examples of Sandpiles. 

The examples are accessible by typing ``sandpiles.NAME``, where 

``NAME`` is the name of the example. You can get a list by typing 

``sandpiles.`` and hitting the TAB key:: 

sandpiles.Complete 

sandpiles.Cycle 

sandpiles.Diamond 

sandpiles.Grid 

sandpiles.House 

See the documentation for each particular type of example for full details. 

""" 

from __future__ import print_function 

from sage.sandpiles.sandpile import Sandpile 

from sage.graphs.graph_generators import graphs 

class SandpileExamples(object): 

""" 

Some examples of sandpiles. 

Here are the available examples; you can also type 

``sandpiles.`` and hit tab to get a list: 

- :meth:`Complete` 

- :meth:`Cycle` 

- :meth:`Diamond` 

- :meth:`Grid` 

- :meth:`House` 

EXAMPLES:: 

sage: s = sandpiles.Complete(4) 

sage: s.invariant_factors() 

[1, 4, 4] 

sage: s.laplacian() 

[ 3 -1 -1 -1] 

[-1 3 -1 -1] 

[-1 -1 3 -1] 

[-1 -1 -1 3] 

""" 

def __call__(self): 

r""" 

If sandpiles() is executed, return a helpful message. 

INPUT: 

None 

OUTPUT: 

None 

EXAMPLES:: 

sage: sandpiles() 

Try sandpiles.FOO() where FOO is in the list: 

<BLANKLINE> 

Complete, Cycle, Diamond, Fan, Grid, House, Wheel 

""" 

print('Try sandpiles.FOO() where FOO is in the list:\n') 

print(" " + ", ".join([str(i) for i in dir(sandpiles) 

if i[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) 

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 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 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 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 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 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) 

sandpiles = SandpileExamples()