Using Algebraic Properties of Minimal Idempotents for Exhaustive Computer Generation of Association Schemes During the past few years we have obtained several new computer classification results on association schemes and in particular distance regular and strongly regular graphs. Central to our success is the use of two algebraic constraints based on properties of the minimal idempotents E_i of these association schemes : the fact that they are positive semidefinite and that they have known rank.   Incorporating these constraints into an actual isomorphfree exhaustive generation algorithm turns out to be somewhat complicated in practice. The main problem to be solved is that of numerical inaccuracy: we do not want to discard a potential solution because a value which is close to zero is misinterpreted as being negative (in the first case) or nonzero (in the second).   In this paper we give details on how this can be accomplished and also list some new classification results that have been recently obtained using this technique: the uniqueness of the strongly regular (126,50,13,24) graph and some new examples of antipodal distance regular graphs. We give an explicit description of a new antipodal distance regular 3cover of K_14, with vertices that can be represented as ordered triples of collinear points of the Fano plane.

The Lexicographic Method for the Threshold Cover Problem The lexicographic method is a technique that was introduced by Hell and Huang Journal of Graph Theory, 20(3): 361xe2x80x93374, 1995 as a way to simplify the problems of recognizing and obtaining representations of comparability graphs, proper circulararc graphs and proper interval graphs. This method gives rise to conceptually simple recognition algorithms and leads to much simpler proofs for some characterization theorems for these classes. Threshold graphs are a class of graphs that have many equivalent definitions and have applications in integer programming and set packing problems. A graph is said to have a threshold cover of size k if its edges can be covered using k threshold graphs. Chvatal and Hammer conjectured in 1977 that given a graph G, a suitably constructed auxiliary graph (Gu0027) has chromatic number equal to the minimum size of a threshold cover of G. Although this conjecture was shown to be false in the general case by Cozzens and Leibowitz, it was shown to be true for graphs having a threshold cover of size 2 by Raschle and Simon Proceedings of the Twentyseventh Annual ACM Symposium on Theory of Computing, STOC xe2x80x9995, pages 650xe2x80x93661, 1995. That is, a graph G has a threshold cover of size 2 if and only if (Gu0027) is bipartitexe2x80x94this is the only known forbidden structure characterization of graphs having a threshold cover of size 2. We show how the lexicographic method can be used to obtain a completely new and much simpler proof for this result. This method also gives rise to a simple new LexBFSbased algorithm for recognizing graphs having a threshold cover of size 2. Although this algorithm is not the fastest known, it is a certifying algorithm that matches the time complexity of the fastest known certifying algorithm for this problem. The algorithm can also be easily adapted to give a certifying recognition algorithm for bipartite graphs that can be covered by two chain subgraphs.