def LCF_graph(n,shift_list,repeats):
"""
Return the cubic graph specified in LCF notation.
LCF notation (LCF=Lederberg-Coxeter-Fruchte) is a compressed
notation used in the generation of various cubic Hamiltonian
graphs of high symmetry. See, for example, dodecahedral_graph,
desargues_graph, heawood_graph and pappus_graph below.
n (number of nodes)
The starting graph is the n-cycle with nodes 0,...,n-1.
(The null graph is returned if n < 0.)
shift_list = [s1,s2,..,sk], a list of integer shifts mod n,
repeats
integer specifying the number of times that shifts in shift_list
are successively applied to each v_current in the n-cycle
to generate an edge between v_current and v_current+shift mod n.
For v1 cycling through the n-cycle a total of k*repeats
with shift cycling through shiftlist repeats times connect
v1 with v1+shift mod n
The utility graph K_{3,3}
>>> G=nx.LCF_graph(6,[3,-3],3)
The Heawood graph
>>> G=nx.LCF_graph(14,[5,-5],7)
See http://mathworld.wolfram.com/LCFNotation.html for a description
and references.
"""
if n <= 0:
return empty_graph()
# start with the n-cycle
G=cycle_graph(n)
G.name="LCF_graph"
nodes=G.nodes()
n_extra_edges=repeats*len(shift_list)
# edges are added n_extra_edges times
# (not all of these need be new)
if n_extra_edges < 1:
return G
for i in range(n_extra_edges):
shift=shift_list[i%len(shift_list)] #cycle through shift_list
v1=nodes[i%n] # cycle repeatedly through nodes
v2=nodes[(i + shift)%n]
G.add_edge(v1, v2)
return G
def LCF_graph(n, shift_list, repeats):
"""
Return the cubic graph specified in LCF notation.
LCF notation (LCF=Lederberg-Coxeter-Fruchte) is a compressed
notation used in the generation of various cubic Hamiltonian
graphs of high symmetry. See, for example, dodecahedral_graph,
desargues_graph, heawood_graph and pappus_graph below.
n (number of nodes)
The starting graph is the n-cycle with nodes 0,...,n-1.
(The null graph is returned if n < 0.)
shift_list = [s1,s2,..,sk], a list of integer shifts mod n,
repeats
integer specifying the number of times that shifts in shift_list
are successively applied to each v_current in the n-cycle
to generate an edge between v_current and v_current+shift mod n.
For v1 cycling through the n-cycle a total of k*repeats
with shift cycling through shiftlist repeats times connect
v1 with v1+shift mod n
The utility graph K_{3,3}
>>> G=nx.LCF_graph(6,[3,-3],3)
The Heawood graph
>>> G=nx.LCF_graph(14,[5,-5],7)
See http://mathworld.wolfram.com/LCFNotation.html for a description
and references.
"""
if n <= 0:
return empty_graph()
# start with the n-cycle
G = cycle_graph(n)
G.name = "LCF_graph"
nodes = G.nodes()
n_extra_edges = repeats * len(shift_list)
# edges are added n_extra_edges times
# (not all of these need be new)
if n_extra_edges < 1:
return G
for i in range(n_extra_edges):
shift = shift_list[i % len(shift_list)] #cycle through shift_list
v1 = nodes[i % n] # cycle repeatedly through nodes
v2 = nodes[(i + shift) % n]
G.add_edge(v1, v2)
return G
def frucht_graph():
"""Return the Frucht Graph.
The Frucht Graph is the smallest cubical graph whose
automorphism group consists only of the identity element.
"""
G=cycle_graph(7)
G.add_edges_from([[0,7],[1,7],[2,8],[3,9],[4,9],[5,10],[6,10],
[7,11],[8,11],[8,9],[10,11]])
G.name="Frucht Graph"
return G
def frucht_graph():
"""Return the Frucht Graph.
The Frucht Graph is the smallest cubical graph whose
automorphism group consists only of the identity element.
"""
G = cycle_graph(7)
G.add_edges_from([[0, 7], [1, 7], [2, 8], [3, 9], [4, 9], [5, 10], [6, 10],
[7, 11], [8, 11], [8, 9], [10, 11]])
G.name = "Frucht Graph"
return G
