The Spectra of Complementary Subgraphs in a Strongly Regular Graph

As an application of Jacobiʹs identity on complementary minors, one can exhibit a simple explicit relation between the characteristic polynomials of any pair of complementary induced subgraphs in a strongly regular graph. In particular, there is an explicit relation between the spectra of the first and second subconstituents with respect to any vertex. Several consequences will be presented; for example, there follow rather strong parametric restrictions on an SRG that has some second subconstituent isomorphic to an antipodal distance-regular graph of diameter three. The latter class of graphs also admits a palatable complementary-minors formula; this will be stated, together with an application.