Independent Sets In Association Schemes

Abstract

Let X be k-regular graph on v vertices and let τ denote the least eigenvalue of its adjacency matrix A(X). If α(X) denotes the maximum size of an independent set in X, we have the following well known bound:It is less well known that if equality holds here and S is a maximum independent set in X with characteristic vector x, then the vectoris an eigenvector for A(X) with eigenvalue τ . In this paper we show how this can be used to characterise the maximal independent sets in certain classes of graphs. As a corollary we show that a graph defined on the partitions of {1, . . . , 9} with three cells of size three is a core. * Research supported by NSERC.

Multiplicity-Free Permutation Representations of the Symmetric Group

“…(e) A n−k × G k ≤ H ≤ S n−k × G k where k = 5, 6 or 9 and G k is AGL(1, 5), P GL(2, 5) or P ΓL (2,8) respectively.…”

“…Beaumont and Peterson [2] have determined the subgroups of S n that are t-homogeneous for all t ≤ n. They are S n , A n , AGL(1, 5) with n = 5, P GL(2, 5) with n = 6, and P GL (2,8) or P ΓL(2, 8) with n = 9.…”

“…From Corollary 2.5 there are four cases to consider, first where ℓ = 5 and stab G (B i ) = AGL(1, 5), second where ℓ = 6 and stab G (B i ) = P GL (1,5) and finally where ℓ = 9 and stab G (B i ) = P GL (2,8) or P ΓL (2,8).…”

“…For k < 11 the decomposition into irreducible characters for P ΓL(2, 8) × A k can be calculated in GAP. If k ≥ 11 Then the the decomposition of P ΓL(2, 8) × A k is the sum of the partitions formed by adding k boxes to the partitions in the decomposition of of P ΓL (2,8) first, in such a way that no two boxes are in the same row and second in such a way that no two boxes are in the same column. The decomposition of P ΓL(2, 8) is it is tedious but not difficult to see that for k ≥ 11 each of these partitions is distinct.…”

“…Since S k ≀S k is not multiplicity free when k > 3, the method of Godsil and Newman [8] will not extend directly.…”