Counting substructures and eigenvalues I: triangles

Abstract

Motivated by the counting results for color-critical subgraphs by Mubayi [Adv. Math., 2010], we study the phenomenon behind Mubayi's theorem from a spectral perspective and start up this problem with the fundamental case of triangles. We prove tight bounds on the number of copies of triangles in a graph with a prescribed number of vertices and edges and spectral radius. Let $n$ and $m$ be the order and size of a graph. Our results extend those of Nosal, who proved there is one triangle if the spectral radius is more than $\sqrt{m}$, and of Rademacher, who proved there are at least $\lfloor\frac{n}{2}\rfloor$ triangles if the number of edges is more than that of 2-partite Tur\'an graph. These results, together with two spectral inequalities due to Bollob\'as and Nikiforov, can be seen as a solution to the case of triangles of a problem of finding spectral versions of Mubayi's theorem. In addition, we give a short proof of the following inequality due to Bollob\'as and Nikiforov [J. Combin. Theory Ser. B, 2007]: $t(G)\geq \frac{\lambda(G)(\lambda^2(G)-m)}{3}$ and characterize the extremal graphs. Some problems are proposed in the end.