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occupied. However, if occupied, they function as hubs linking together up to
t
u
2-clusters. The sizes of the groups are
|
S
1
|
=
|
S
2
|
/
2=
|
S
3
|
. For illustration see
Fig. 3.
Fig. 3.
A typical pattern found for
I
= 10, and similarly for
I
= 60. The occupied
vertices form 2-clusters, some of which are interlinked via hubs. The vertices are labeled
with the decimal expression of their bitstring. The sum of the indices within a 2-cluster
is always 6207 in this 2-cluster configuration. The determinant bit positions are 7 and
12. Figure produced using yEd [18].
Looking at the vertex indices
i
v
in decimal representation we made a sur-
prising observation: The sum of the two indices in a 2-cluster is constant in the
whole graph.
A look at the bitstrings of the nodes of all 2-clusters revealed, that they are
identical in exactly two bits, say at position
k
and
l
. The remaining
d
2bit
positions assume all 2
d−
2
possible values. Inside a cluster the two bitstrings are
complementary in these positions. Thus, the 2-clusters have a two-mismatch link
and we write symbolically
−
···
b
k
···
b
l
···
···
b
k
···
b
l
···
,
connects to
(1)
where the bar denotes the bit inversion.
The other groups,
S
2
and
S
3
, have similar structural properties. The bitstrings
of all stable holes are also equal in the same two bit positions
k
and
l
. However,
they are inverse to
b
k
and
b
l
of the occupied vertices. Potential hubs have exactly