A weighted association scheme is a scheme with an edge weight function, which for our purposes will take values ±1. When the scheme has a coherent fusiona merging of classes resulting in another association scheme-the edge weights on the fusion scheme are inherited. The reverse process involves the coherent closure of a weighted scheme: the smallest coherent algebra containing the weighted adjacency matrices. The weight function applied to this closure is necessarily trivial, meaning constant on classes of the associated configuration. In this work there are two main objects of study: minimal rank coherent closures of strongly regular graphs with regular weights; and strongly regular graphs with regular weights which are obtained as fusions of association schemes with trivial regular weights. Both of these extend work of D. Taylor on regular two-graphs and their interactions with strongly regular graphs. We obtain regular weights on strongly regular graphs with 125 and 256 vertices as fusions of rank 5 weighted schemes, and a family of rank 6, primitive, non-metric schemes which are minimal-rank closures of weighted lattice graphs. Our main result is the classification of these coherent closures of rank 4. They are imprimitive rank 4 association schemes, which have been studied generally by E. R. van Dam, and Y. Chang and T. Huang.
We introduce the covering configuration induced by a regular weight defined on a coherent configuration. This construction generalizes the well-known equivalence of regular two-graphs and antipodal double covers of complete graphs. It also recovers, as special cases, the rank 6 association schemes connected with regular 3-graphs, and certain extended Q-bipartite doubles of cometric association schemes. We articulate sufficient conditions on the parameters of a coherent configuration for it to arise as a covering configuration.
Type-II matrices are nonzero complex matrices that were introduced in connection with spin models for link invariants. Type-II matrices have been found in connection with symmetric designs, sets of equiangular lines, strongly regular graphs, and some distance regular graphs. We investigate weighted complete and strongly regular graphs, and show that type-II matrices arise in this setting as well.
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