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- W4225574097 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 $lfloorfrac{n}{2}rfloor$ 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." @default.
- W4225574097 created "2022-05-05" @default.
- W4225574097 creator A5033679601 @default.
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- W4225574097 date "2021-12-23" @default.
- W4225574097 modified "2023-09-25" @default.
- W4225574097 title "Counting substructures and eigenvalues I: triangles" @default.
- W4225574097 doi "https://doi.org/10.48550/arxiv.2112.12937" @default.
- W4225574097 hasPublicationYear "2021" @default.
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