A classical result of Erd\H{o}s 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 Erd\H{o}s--Rademacher's theorem. Moreover, Mubayi [Adv. Math. 225 (2010)] extends the result of Erd\H{o}s 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.

Yongtao Li

Lu Lu

Yuejian Peng

Advances in Applied Mathematics

跨学科的应用数学进展致力于发表关于应用数学严格方法和结果的原创文章和调查文章。该杂志的特色文章涉及离散数学、离散概率论、理论统计学、数学生物学和生物信息学、应用交换代数和代数几何、凸性理论、实验数学、理论计算机科学和其他领域。该杂志是数学家、计算机科学家、应用数学家、物理学家、统计学家和生物学家获取当前信息的绝佳来源。在过去十年中，《应用数学进展》发表了由我们这个时代许多最重要的数学家撰写的研究论文。

Advances in Applied Mathematics is an interdisciplinary journal dedicated to publishing original research and survey articles on applied mathematics. The featured articles cover a wide range of topics, including discrete mathematics, discrete probability theory, theoretical statistics, mathematical biology and bioinformatics, applied algebraic geometry, convexity theory, experimental mathematics, theoretical computer science, and more. It serves as an essential resource for mathematicians, computer scientists, applied mathematicians, physicists, statisticians, and biologists to stay updated on current research. Over the past decade, Advances in Applied Mathematics has published research papers authored by many of the most significant mathematicians of our time.

《Advances in Applied Mathematics》是一份跨学科的应用数学期刊，致力于发表关于应用数学的原创研究和综述文章。该期刊的特色文章涵盖离散数学、离散概率论、理论统计学、数学生物学与生物信息学、应用交换代数与代数几何、凸性理论、实验数学、理论计算机科学等多个领域。它是数学家、计算机科学家、应用数学家、物理学家、统计学家和生物学家获取最新研究成果的重要来源。在过去十年中，《Advances in Applied Mathematics》发表了许多当代重要数学家的研究论文。

Category	Quartile
数学	3
数学, 应用数学	3

A spectral Erdős-Rademacher theorem

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Category	Quartile
MATHEMATICS, APPLIED	3