Counting substructures and eigenvalues I: triangles

TL;DR

This paper investigates the number of triangles in graphs using spectral graph theory, extending classical results and providing tight bounds based on spectral radius, edges, and vertices, with implications for Mubayi's theorem.

Contribution

It offers new spectral bounds on triangle counts in graphs, extending prior combinatorial results and characterizing extremal graphs for these bounds.

Findings

Proved tight bounds on triangle counts using spectral radius.

Extended classical results by Nosal and Rademacher.

Provided a spectral inequality relating eigenvalues and triangle counts.

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.