A spectral version of Mantel's theorem

Zhai, Mingqing; Shu, Jinlong

doi:10.1016/j.disc.2021.112630

Cited by 29 publications

(30 citation statements)

References 13 publications

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“…We remark here that λ 2 1 (G) + λ 2 2 (G) ≤ m does not hold for the C 4 -free graphs G. Indeed, take G = K + 1,m−1 , the graph obtained from the star K 1,m−1 by adding an edge into its independent set. For example, setting m = 20, we have λ 1 (K In 2022, Zhai and Shu [54] proved a further improvement on Theorem 1.5. Before stating their result, we need to introduce the extremal graph firstly.…”

Section: The Spectral Extremal Graph Problemsmentioning

confidence: 97%

“…On the other hand, the inequality (3) in Nosal's theorem boosted the great interests of studying the maximum spectral radius of graphs in terms of the number of edges, instead of the number of vertices; see [37] for an extension on K r+1 -free graphs, [39] for an analogue of C 4 -free graphs, [53] for further extensions on K 2,r+1 -free graphs, and similar results of C 5 -free and C 6 -free graphs as well, [42] for an extension on B kfree graphs, where B k denotes the book graph consisting of k triangles sharing a common edge, and [31,54] for refinements on non-bipartite triangle-free graphs. In this paper, we will focus mainly on the extremal spectral problems for graphs with given number of edges, which is becoming increasingly an important and popular topic in recent research on spectral graph theory.…”

Section: The Spectral Extremal Graph Problemsmentioning

confidence: 99%

“…In Section 2, we shall present an alternative proof of Theorem 1.6. 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.…”

mentioning

confidence: 92%

“…So β(m) is also the largest root of H(x). The way that Lin, Ning and Wu [31] proved Theorem 1.5 is original, and the line of the proof of Zhai and Shu [54] for Theorem 1.6 is technical. This paper is organized as follows.…”

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confidence: 99%

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

Li¹,

Peng²

2022

Preprint

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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 states that if G is a triangle-free graph with m edges, then λ(G) ≤ √ 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 trianglefree 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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Section: The Spectral Extremal Graph Problemsmentioning

confidence: 97%

Section: The Spectral Extremal Graph Problemsmentioning

confidence: 99%

mentioning

confidence: 92%

mentioning

confidence: 99%

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

Li¹,

Peng²

2022

Preprint

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“…Furthermore, Lin et al [13] proved that every non-bipartite graph on m edges contains a triangle if λ(G) ≥ √ m − 1 unless G is a C 5 (possibly together with some isolated vertices). Only very recently, Zhai and Shu [25] extended it as follows:…”

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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Note on Mantel Theorem and Turán Theorem

Ning

2024

Graphs and Combinatorics

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A spectral version of Mantel's theorem

Cited by 29 publications

References 13 publications

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

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

Counting substructures and eigenvalues I: triangles

Note on Mantel Theorem and Turán Theorem

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