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 √ m, and of Rademacher, who proved there are at least ⌊ n 2 ⌋ triangles if the number of edges is more than that of 2-partite 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 [J. Combin. Theory Ser. B, 2007]: t(G) ≥ λ(G)(λ 2 (G)−m) 3 and characterize the extremal graphs. Some problems are proposed in the end.

The Maximum Spectral Radius of Non-Bipartite Graphs Forbidding Short Odd Cycles

“…By applying Cauchy's interlacing theorem of all eigenvalues, we will find some forbidden induced subgraphs and refine the structure of the desired extremal graph. A key idea relies on the eigenvalue interlacing theorem and a counting lemma [43], which established the relation between eigenvalues and the number of triangles of a graph.…”

“…The second lemma needed in this paper is a triangle counting lemma in terms of both the eigenvalues and the size of a graph, it could be seen from [43]. This could be viewed as a useful variant of (7) by using n i=1 λ 2 i = tr(A 2 ) = n i=1 d i = 2m.…”

“…The present proof is different from the original proof in [54]. Our proof uses and develops the ideas in both [31] and [43], we shall make use of the information of all eigenvalues of graphs, instead of the second largest eigenvalue only. This proof could introduce the main ideas of the approach of our paper, without some technicalities that arise in the other cases, i.e., it can help us to deal with the extremal spectral problem for graphs without short odd cycles.…”

“…Theorem 3 asserts that if G is a graph with λ(G) √ m, then either G contains a triangle, or G is a complete bipartite graph. Very recently, Ning and Zhai [43] proved an elegant spectral counting result, which states that if G is an m-edge graph with λ(G) √ m, then G has at least…”

The maximum spectral radius of non-bipartite graphs forbidding short odd cycles

“…The Nosal Theorem 1.3 asserts that if G is a graph with λ(G) ≥ √ m, then either G contains a triangle, or G is a complete bipartite graph. Very recently, Ning and Zhai [43] proved an elegant spectral counting result, which states that if G is an m-edge graph with λ(G) ≥ √ m, then G has at least…”