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. Collections: Mathematics