Vertex-Switching Reconstruction and Folded Cubes

In this note we use eigenvalues of folded cubes to simplify an analogue of Kellyʹs Lemma for vertex-switching reconstruction due to Krasikov and Roditty. Our new version states that the number of subgraphs (or induced subgraphs) of ann-vertex graphGisomorphic to a givenm-vertex graph can be found from thes-vertex-switching deck ofGprovided the Krawtchouk polynomialKsnhas no even roots in [0, m]. This generalizes a condition of Stanley fors-vertex-switching reconstructibility. We also comment on the role of cubes and folded cubes in the theory of vertex-switching reconstruction.