A spectral Erdős-Rademacher theorem

A classical result of Erdős and Rademacher (1955) indicates a supersaturation phenomenon. It says that if G is a graph on n vertices with at least $\lfloor n^2/4\rfloor +1$ edges, then G contains at least $\lfloor n/2\rfloor$ triangles. We prove a spectral version of Erdős–Rademacher's theorem. Moreover, Mubayi (2010) [28] extends the result of Erdős and Rademacher from a triangle to any color-critical graph. It is interesting to study the extension of Mubayi from a spectral perspective. However, it is not apparent to measure the increment on the spectral radius of a graph comparing to the traditional edge version (Mubayi's result). In this paper, we provide a way to measure the increment on the spectral radius of a graph and propose a spectral version on the counting problems for color-critical graphs.