Counting substructures and eigenvalues I: Triangles

Abstract

Motivated by the counting results for color-critical subgraphs by Mubayi (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 for the number of copies of triangle in a graph of order n and size m with spectral radius λ. Our results extend those of Nosal, who proved there is one triangle if λ>m, and of Rademacher, who proved there are at least ⌊n2⌋ triangles if the size is more than that of bipartite Turán graph. These results, together with two spectral inequalities due to Bollobás 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 Bollobás and Nikiforov (2007): the number of copies of triangle t(G)≥λ(λ2−m)3, and characterize the extremal graphs. Some problems are proposed in the end.