Equitable switching and spectra of graphs

Let G be a simple graph and π an equitable partition of the vertex set V(G). It is well known that the characteristic polynomial φ(G/π;λ) of a quotient graph G/π divides that of G. The Seidel swiching Gσ of G with respect to a proper subset V of V(G) is called equitableif V is a union of some cells in π. We show that if Gσ is equitable, φ(G;λ)/φ(G/π;λ)=φ(Gσ;λ)/φ(Gσ/π;λ). Some applications of this theorem to generalized composition graphs, isospectral graphs, integral graphs and Ramanujan graphs are contained.