Let G be a graph with m edges and spectral radius λ 1 . Let bk (G) stand for the maximal number of triangles with a common edge in G.In 1970 Nosal proved that if λ 2 1 > m, then G contains a triangle. In this paper we show that the same premise implies thatThis result settles a conjecture of Zhai, Lin, and Shu. Write λ 2 for the second largest eigenvalue of G. Recently, Lin, Ning, and Wu showed that if G is a triangle-free graph of order at least three, thenthereby settling the simplest case of a conjecture of Bollobás and the author. We give a simpler proof of their result.