On the components of NEPS of connected bipartite graphs

Abstract

Back in 1983, D. Cvetković posed the conjecture that the components of NEPS of connected bipartite graphs are almost cospectral. In 2000, we showed that this conjecture does not hold for infinitely many bases of NEPS, and we posed a necessary condition on the base of NEPS for NEPS to have almost cospectral components. At the same time, D. Cvetković posed weaker version of his original conjecture which claims that each eigenvalue of NEPS is also the eigenvalue of each component of NEPS. Here we prove this weaker conjecture, give an upper bound on the multiplicity of an eigenvalue of NEPS as an eigenvalue of its component, give new sufficient condition for the almost cospectrality of components of NEPS of connected bipartite graphs, and characterize the bases of NEPS which satisfy this condition.