A spectral Erdős-Rademacher theorem

A classical result of Erd6s and Rademacher (1955) indicates a supersaturation phenomenon. It says that if G is a graph on n vertices with at least L n 2 / 4 ] + 1 edges, then G contains at least L n/ 2] triangles. We prove a spectral version of Erd6s- Rademacher's theorem. Moreover, Mubayi (2010) [28] extends the result of Erd6s 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. (c) 2024 Elsevier Inc. All rights reserved.