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Eigenvalues of the Partition Graphs November 23, 2009
 

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 k-partition of an n-set
is a partition of an n-set into k classes each of size . If k does not divide
n, it is not possible to have uniform k-partitions of an n-set. In this case,
almost-uniform partitions are considered. For positive integers n, k, with
n = k + r where 0 r < k, an almost-uniform k-partition of an n-set is a
partition of an n-set 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 k-partitions of an n-set 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 almost-uniform k-partitions of an
n-set. Vertices are adjacent if and only if the corresponding partitions are
qualitatively independent.

  

Source: Argerami, Martin - Department of Mathematics and Statistics, University of Regina

 

Collections: Mathematics