<s>
In	O
the	O
mathematical	O
field	O
of	O
graph	O
theory	O
,	O
the	O
Fibonacci	B-Algorithm
cubes	I-Algorithm
or	O
Fibonacci	O
networks	O
are	O
a	O
family	O
of	O
undirected	O
graphs	O
with	O
rich	O
recursive	O
properties	O
derived	O
from	O
its	O
origin	O
in	O
number	O
theory	O
.	O
</s>
<s>
Mathematically	O
they	O
are	O
similar	O
to	O
the	O
hypercube	O
graphs	O
,	O
but	O
with	O
a	O
Fibonacci	B-Algorithm
number	I-Algorithm
of	O
vertices	O
.	O
</s>
<s>
Fibonacci	B-Algorithm
cubes	I-Algorithm
were	O
first	O
explicitly	O
defined	O
in	O
in	O
the	O
context	O
of	O
interconnection	O
topologies	B-Architecture
for	O
connecting	O
parallel	O
or	O
distributed	O
systems	O
.	O
</s>
<s>
The	O
Fibonacci	B-Algorithm
cube	I-Algorithm
may	O
be	O
defined	O
in	O
terms	O
of	O
Fibonacci	B-Algorithm
codes	I-Algorithm
and	O
Hamming	O
distance	O
,	O
independent	O
sets	O
of	O
vertices	O
in	O
path	O
graphs	O
,	O
or	O
via	O
distributive	O
lattices	O
.	O
</s>
<s>
Like	O
the	O
hypercube	O
graph	O
,	O
the	O
vertices	O
of	O
the	O
Fibonacci	B-Algorithm
cube	I-Algorithm
of	O
order	O
n	O
may	O
be	O
labeled	O
with	O
bitstrings	B-Data_Structure
of	O
length	O
n	O
,	O
in	O
such	O
a	O
way	O
that	O
two	O
vertices	O
are	O
adjacent	O
whenever	O
their	O
labels	O
differ	O
in	O
a	O
single	O
bit	O
.	O
</s>
<s>
However	O
,	O
in	O
a	O
Fibonacci	B-Algorithm
cube	I-Algorithm
,	O
the	O
only	O
labels	O
that	O
are	O
allowed	O
are	O
bitstrings	B-Data_Structure
with	O
no	O
two	O
consecutive	O
1	O
bits	O
.	O
</s>
<s>
If	O
the	O
labels	O
of	O
the	O
hypercube	O
are	O
interpreted	O
as	O
binary	O
numbers	O
,	O
the	O
labels	O
in	O
the	O
Fibonacci	B-Algorithm
cube	I-Algorithm
are	O
a	O
subset	O
,	O
the	O
fibbinary	O
numbers	O
.	O
</s>
<s>
There	O
are	O
Fn	O
+	O
2	O
labels	O
possible	O
,	O
where	O
Fn	O
denotes	O
the	O
nth	O
Fibonacci	B-Algorithm
number	I-Algorithm
,	O
and	O
therefore	O
there	O
are	O
Fn	O
+	O
2	O
vertices	O
in	O
the	O
Fibonacci	B-Algorithm
cube	I-Algorithm
of	O
order	O
n	O
.	O
</s>
<s>
The	O
nodes	O
of	O
such	O
a	O
network	O
may	O
be	O
assigned	O
consecutive	O
integers	O
from	O
0	O
to	O
Fn	O
+	O
21	O
;	O
the	O
bitstrings	B-Data_Structure
corresponding	O
to	O
these	O
numbers	O
are	O
given	O
by	O
their	O
Zeckendorf	B-Algorithm
representations	I-Algorithm
.	O
</s>
<s>
The	O
Fibonacci	B-Algorithm
cube	I-Algorithm
of	O
order	O
n	O
is	O
the	O
simplex	O
graph	O
of	O
the	O
complement	O
graph	O
of	O
an	O
n-vertex	O
path	O
graph	O
.	O
</s>
<s>
That	O
is	O
,	O
each	O
vertex	O
in	O
the	O
Fibonacci	B-Algorithm
cube	I-Algorithm
represents	O
a	O
clique	O
in	O
the	O
path	O
complement	O
graph	O
,	O
or	O
equivalently	O
an	O
independent	O
set	O
in	O
the	O
path	O
itself	O
;	O
two	O
Fibonacci	B-Algorithm
cube	I-Algorithm
vertices	O
are	O
adjacent	O
if	O
the	O
cliques	O
or	O
independent	O
sets	O
that	O
they	O
represent	O
differ	O
by	O
the	O
addition	O
or	O
removal	O
of	O
a	O
single	O
element	O
.	O
</s>
<s>
Therefore	O
,	O
like	O
other	O
simplex	O
graphs	O
,	O
Fibonacci	B-Algorithm
cubes	I-Algorithm
are	O
median	O
graphs	O
and	O
more	O
generally	O
partial	O
cubes	O
.	O
</s>
<s>
The	O
median	O
of	O
any	O
three	O
vertices	O
in	O
a	O
Fibonacci	B-Algorithm
cube	I-Algorithm
may	O
be	O
found	O
by	O
computing	O
the	O
bitwise	O
majority	O
function	O
of	O
the	O
three	O
labels	O
;	O
if	O
each	O
of	O
the	O
three	O
labels	O
has	O
no	O
two	O
consecutive	O
1	O
bits	O
,	O
the	O
same	O
is	O
true	O
of	O
their	O
majority	O
.	O
</s>
<s>
The	O
Fibonacci	B-Algorithm
cube	I-Algorithm
is	O
also	O
the	O
graph	O
of	O
a	O
distributive	O
lattice	O
that	O
may	O
be	O
obtained	O
via	O
Birkhoff	O
's	O
representation	O
theorem	O
from	O
a	O
zigzag	O
poset	O
,	O
a	O
partially	O
ordered	O
set	O
defined	O
by	O
an	O
alternating	O
sequence	O
of	O
order	O
relations	O
a	O
b	O
c	O
d	O
e	O
f	O
...	O
</s>
<s>
There	O
is	O
also	O
an	O
alternative	O
graph-theoretic	O
description	O
of	O
the	O
same	O
lattice	O
:	O
the	O
independent	O
sets	O
of	O
any	O
bipartite	O
graph	O
may	O
be	O
given	O
a	O
partial	O
order	O
in	O
which	O
one	O
independent	O
set	O
is	O
less	O
than	O
another	O
if	O
they	O
differ	O
by	O
removing	O
elements	O
from	O
one	O
side	O
of	O
the	O
bipartition	O
and	O
adding	O
elements	O
to	O
the	O
other	O
side	O
of	O
the	O
bipartition	O
;	O
with	O
this	O
order	O
,	O
the	O
independent	O
sets	O
form	O
a	O
distributive	O
lattice	O
,	O
and	O
applying	O
this	O
construction	O
to	O
a	O
path	O
graph	O
results	O
in	O
the	O
lattice	O
associated	O
with	O
the	O
Fibonacci	B-Algorithm
cube	I-Algorithm
.	O
</s>
<s>
The	O
Fibonacci	B-Algorithm
cube	I-Algorithm
of	O
order	O
n	O
may	O
be	O
partitioned	O
into	O
a	O
Fibonacci	B-Algorithm
cube	I-Algorithm
of	O
order	O
n1	O
(	O
the	O
nodes	O
with	O
labels	O
beginning	O
with	O
a	O
0	O
bit	O
)	O
and	O
a	O
Fibonacci	B-Algorithm
cube	I-Algorithm
of	O
order	O
n2	O
(	O
the	O
nodes	O
with	O
labels	O
beginning	O
with	O
a	O
1	O
bit	O
)	O
.	O
</s>
<s>
Every	O
Fibonacci	B-Algorithm
cube	I-Algorithm
has	O
a	O
Hamiltonian	O
path	O
.	O
</s>
<s>
Thus	O
,	O
e.g.	O
,	O
in	O
the	O
Fibonacci	B-Algorithm
cube	I-Algorithm
of	O
order	O
4	O
,	O
the	O
sequence	O
constructed	O
in	O
this	O
way	O
is	O
(	O
0100-0101-0001-0000-0010	O
)	O
-(1010-1000-1001 )	O
,	O
where	O
the	O
parentheses	O
demark	O
the	O
subsequences	O
within	O
the	O
two	O
subgraphs	O
of	O
the	O
partition	O
.	O
</s>
<s>
Fibonacci	B-Algorithm
cubes	I-Algorithm
with	O
an	O
even	O
number	O
of	O
nodes	O
greater	O
than	O
two	O
have	O
a	O
Hamiltonian	O
cycle	O
.	O
</s>
<s>
investigate	O
the	O
radius	O
and	O
independence	O
number	O
of	O
Fibonacci	B-Algorithm
cubes	I-Algorithm
.	O
</s>
<s>
The	O
diameter	O
of	O
a	O
Fibonacci	B-Algorithm
cube	I-Algorithm
of	O
order	O
n	O
is	O
n	O
,	O
and	O
its	O
radius	O
is	O
n/2	O
(	O
again	O
,	O
rounded	O
up	O
to	O
the	O
nearest	O
integer	O
)	O
.	O
</s>
<s>
showed	O
that	O
it	O
is	O
possible	O
to	O
test	O
whether	O
a	O
graph	O
is	O
a	O
Fibonacci	B-Algorithm
cube	I-Algorithm
in	O
time	O
near-linear	O
in	O
its	O
size	O
.	O
</s>
<s>
and	O
suggested	O
using	O
Fibonacci	B-Algorithm
cubes	I-Algorithm
as	O
a	O
network	B-Architecture
topology	I-Architecture
in	O
parallel	B-Operating_System
computing	I-Operating_System
.	O
</s>
<s>
As	O
a	O
communications	O
network	O
,	O
the	O
Fibonacci	B-Algorithm
cube	I-Algorithm
has	O
beneficial	O
properties	O
similar	O
to	O
those	O
of	O
the	O
hypercube	O
:	O
the	O
number	O
of	O
incident	O
edges	O
per	O
vertex	O
is	O
at	O
most	O
n/2	O
and	O
the	O
diameter	O
of	O
the	O
network	O
is	O
at	O
most	O
n	O
,	O
both	O
proportional	O
to	O
the	O
logarithm	O
of	O
the	O
number	O
of	O
vertices	O
,	O
and	O
the	O
ability	O
of	O
the	O
network	O
to	O
be	O
partitioned	O
into	O
smaller	O
networks	O
of	O
the	O
same	O
type	O
allows	O
it	O
to	O
be	O
split	O
among	O
multiple	O
parallel	B-Operating_System
computation	I-Operating_System
tasks	O
.	O
</s>
<s>
Fibonacci	B-Algorithm
cubes	I-Algorithm
also	O
support	O
efficient	O
protocols	O
for	O
routing	B-Protocol
and	O
broadcasting	B-Operating_System
in	O
distributed	O
computations	O
.	O
</s>
<s>
apply	O
Fibonacci	B-Algorithm
cubes	I-Algorithm
in	O
chemical	O
graph	O
theory	O
as	O
a	O
description	O
of	O
the	O
family	O
of	O
perfect	O
matchings	O
of	O
certain	O
molecular	O
graphs	O
.	O
</s>
<s>
More	O
generally	O
described	O
the	O
class	O
of	O
planar	O
bipartite	O
graphs	O
that	O
have	O
Fibonacci	B-Algorithm
cubes	I-Algorithm
as	O
their	O
resonance	O
graphs	O
.	O
</s>
<s>
Generalized	O
Fibonacci	B-Algorithm
cubes	I-Algorithm
were	O
presented	O
by	O
based	O
on	O
the	O
k-th	O
order	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
,	O
which	O
were	O
later	O
further	O
extended	O
to	O
a	O
larger	O
class	O
of	O
networks	O
called	O
the	O
Linear	O
Recursive	O
Networks	O
by	O
based	O
on	O
more	O
general	O
forms	O
of	O
linear	O
recursions	O
.	O
</s>
<s>
modified	O
the	O
second	O
order	O
Fibonacci	B-Algorithm
cubes	I-Algorithm
based	O
on	O
different	O
initial	O
conditions	O
.	O
</s>
<s>
Another	O
related	O
graph	O
is	O
the	O
Lucas	O
cube	O
,	O
a	O
graph	O
with	O
a	O
Lucas	B-Algorithm
number	I-Algorithm
of	O
vertices	O
defined	O
from	O
the	O
Fibonacci	B-Algorithm
cube	I-Algorithm
by	O
forbidding	O
a	O
1	O
bit	O
in	O
both	O
the	O
first	O
and	O
last	O
positions	O
of	O
each	O
bitstring	B-Data_Structure
;	O
investigated	O
the	O
coloring	O
properties	O
of	O
both	O
Fibonacci	B-Algorithm
cubes	I-Algorithm
and	O
Lucas	O
cubes	O
.	O
</s>
