3-ranks for strongly regular graphs

Abstract

In the study of strongly regular graphs, ranks of adjacency matrices (Laplacians actually) are extensively used to demonstrate inequivalence of graphs. Constructions have been given for several families of graphs. Formulas for the ranks in these families are an important tool for understanding their properties. The first and computational challenge is to compute rank modulo 3 of some very large matrices. To our advantage is that the ranks are expected to be relatively small. Typically in these families, the matrix dimension is 3k while the rank modulo 3 is in the vicinity of 2k. Here we discuss a high performance parallel solution to the problem. It involves parallelism at three levels: word-level vectorization of field elements, shared-memory multi-core, and a multi-node distributed memory and file-system modulated level. The implementation has been applied to the case k = 16, wherein the matrix contains approximately 1.85 peta-entries. The second challenge is to discern a formula for the sequence of ranks in a given graph family. Our computations provide further evidence for an existing conjecture concerning the Dickson family of strongly regular graphs and provide a starting point towards finding a formula for the Ding-Yuan and Cohen-Ganley families of graphs.