Counting substructures and eigenvalues I: Triangles

Motivated by the counting results for color-critical subgraphs Mubayi (2010), we study phenomenon behind Mubayi’s theorem from a spectral perspective and start up this problem with fundamental case of triangles. We prove tight bounds number copies triangle in graph order n size m radius λ. Our extend those Nosal, who proved there is one if λ>m, Rademacher, are at least ⌊n2⌋ triangles more than that bipartite Turán graph. These results, together two inequalities due to Bollobás Nikiforov, can be seen as solution finding versions theorem. In addition, give short proof following inequality Nikiforov (2007): t(G)≥λ(λ2−m)3, characterize extremal graphs. Some problems proposed end.