 
Summary: Eigenvalues of the Partition Graphs
November 23, 2009
1 Bound for Max Clique in the Partition Graph
For positive integers n, k, with n = k , a uniform kpartition of an nset
is a partition of an nset into k classes each of size . If k does not divide
n, it is not possible to have uniform kpartitions of an nset. In this case,
almostuniform partitions are considered. For positive integers n, k, with
n = k + r where 0 r < k, an almostuniform kpartition of an nset is a
partition of an nset into k classes, each of size or + 1.
Partitions P = {P1, P2, ..., Pk} and Q = {Q1, Q2, ..., Qk} are called quali
tatively independent if for all i, j {1, ..., k}
Pi Qj = .
If P and Q are qualitatively independent kpartitions of an nset then the
characteristic vectors of P and Q could be two rows in a covering array with
parameters CA(n, b, k).
1.1 Definition (Partition Graph). Let n, k, be positive integers such that
n = k + r where 0 r < k . The partition graph P(n, k) is the
graph whose vertex set is the set of all almostuniform kpartitions of an
nset. Vertices are adjacent if and only if the corresponding partitions are
qualitatively independent.
