Combinatorial Designs, Difference Sets, and Bent Functions as Perfect Colorings of Graphs and Multigraphs

We prove that (1): the characteristic function of each independent set in each regular graph attaining the Delsarte–Hoffman bound is a perfect coloring; (2): each transversal in a uniform regular hypergraph is an independent set in the vertex adjacency multigraph of a hypergraph attaining the Delsarte–Hoffman bound for this multigraph; and (3): the combinatorial designs with parameters - and their -analogs, difference sets, Hadamard matrices, and bent functions are equivalent to perfect colorings of some graphs of multigraphs, in particular, the Johnson graph for - -designs and the Grassmann graph for bent functions.