Root systems and optimal block designs

Abstract

Motivated by a question of C.-S. Cheng on optimal block designs, this paper describes the symmetric matrices with entries 0, +1 and −1, zero diagonal, least eigenvalue strictly greater than −2, and constant row sum. I also describe briefly the motivation for the question. Let A be a symmetric n× n matrix with entries 0, +1 and −1, with zero diagonal and constant row sums, having least eigenvalue greater than −2. The aim of this paper is to describe such matrices. Of course, we may assume that the matrix is “connected”, that is, not permutation-equivalent to one of the form ( B O O C ) . We also note that the constant row sum c is an eigenvalue, and so c ≥ −1. Such a matrix is represented by a set of vectors in a spherical root system, and hence (apart from finitely many examples represented in the exceptional root systems E6, E7 and E8) by either a tree with oriented edges, or a unicyclic graph with edges either signed or oriented. We give a test for recognising when such a graph represents a matrix satisying the conditions of the question. There are many examples. All matrices occurring in the exceptional root systems are determined. 1 Least eigenvalue −1 As a warmup, I consider the case where the least eigenvalue is −1.