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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
 
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