On a theorem of Nosal

Nikiforov, V.

doi:10.48550/arxiv.2104.12171

Cited by 8 publications

(25 citation statements)

References 2 publications

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“…unless it is a complete bipartite graph with possibly some isolated vertices (see [17], this confirmed a conjecture in [22]).…”

Section: Nikiforov's Deleting Small Eigenvalue Edge Methodssupporting

confidence: 54%

“…Let B r be an r-book, that is, the graph obtained from K 2,r by adding one edge within the partition set of two vertices. Very recently, Nikiforov [17] proved that, if m ≥ (12r) 4 and λ(G) ≥ √ m, then G contains a copy of B r+1 , unless G is a complete bipartite graph (possibly with some isolated vertices). This result further extends above two theorems and solves a conjecture proposed in [22].…”

Section: Theorem 2 ([22]mentioning

confidence: 99%

“…Proof of Theorem 3. Let G be a graph with e(G) = m and λ(G) ≥ √ m. By using the Nikiforov DESS Method [17], we first construct a sequence of graphs.…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Counting substructures and eigenvalues II: quadrilaterals

Niu¹,

Zhai²

2021

Preprint

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Let G be a graph and λ(G) be the spectral radius of G. A previous result due to Nikiforov [Linear Algebra Appl., 2009] in spectral graph theory asserted that every graph G on m ≥ 10 edges contains a 4to be the minimum number of copies of 4-cycles in such a graph. A consequence of a recent theorem due to Zhai et al. [European J. Combin., 2021] shows that f (m) = Ω(m). In this article, by somewhat different techniques, we prove that f (m) = Θ(m 2 ). We left the solution to lim m→∞ f (m) m 2 as a problem, and also mention other ones for further study.

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“…unless it is a complete bipartite graph with possibly some isolated vertices (see [17], this confirmed a conjecture in [22]).…”

Section: Nikiforov's Deleting Small Eigenvalue Edge Methodssupporting

confidence: 54%

Section: Theorem 2 ([22]mentioning

confidence: 99%

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Counting substructures and eigenvalues II: quadrilaterals

Niu¹,

Zhai²

2021

Preprint

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“…. For other extensions of Nosal's theorem, see [6,12,19,24]. In 2007, Bollobás and Nikiforov [2] proved a number of relations between the number of cliques of a graph G and λ(G).…”

Section: Introductionmentioning

confidence: 99%

“…With the method similar to the one proving Theorem 6, we shall present a short new proof of Theorem 4. Very recently, Theorem 4 was further improved by Nikiforov [19] to λ 3 − λ • m + cλ • t ′′ ≤ 3t for connected non-bipartite graphs, where [23,Conjecture 5.2]). The original inequality (Theorem 4) is also used as a tool for obtaining a spectral version of extremal number of friendship graphs [10].…”

Section: Introductionmentioning

confidence: 99%

Counting substructures and eigenvalues I: triangles

Niu¹,

Zhai²

2021

Preprint

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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.

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

Li¹,

Peng

2022

Electron. J. Combin.

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It is well-known that eigenvalues of graphs can be used to describe structural properties and parameters of graphs. A theorem of Nosal and Nikiforov states that if $G$ is a triangle-free graph with $m$ edges, then $\lambda (G)\le \sqrt{m}$, equality holds if and only if $G$ is a complete bipartite graph. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] proved a generalization for non-bipartite triangle-free graphs. Moreover, Zhai and Shu [Discrete Math. 345 (2022)] presented a further improvement. In this paper, we present an alternative method for proving the improvement by Zhai and Shu. Furthermore, the method can allow us to give a refinement on the result of Zhai and Shu for non-bipartite graphs without short odd cycles.

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On a theorem of Nosal

Cited by 8 publications

References 2 publications

Counting substructures and eigenvalues II: quadrilaterals

Counting substructures and eigenvalues II: quadrilaterals

Counting substructures and eigenvalues I: triangles

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

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