New proofs of stability theorems on spectral graph problems

Li, Yongtao; Peng, Yuejian

doi:10.48550/arxiv.2203.03142

Cited by 3 publications

(3 citation statements)

References 36 publications

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“…In view of this perspective, various extensions and generalizations on inequality (4) have been obtained in the literature; see, e.g., [51,38,19,25] for extensions on K r+1 -free graphs with given order; see [7,40] for relations between cliques and spectral radius and [41,10,27] for surveys. Very recently, Lin, Ning and Wu [31,Theorem 1.4] proved a generalization on (4) for non-bipartite triangle-free graphs and provided a spectral version of Theorem 2; see [29] for an alternative proof and refinement of spectral Turán theorem, and [28] for more stability theorems on spectral graph problems. In addition, Lin and Guo [32] proved an extension of non-bipartite graphs without short odd cycles.…”

Section: The Spectral Extremal Graph Problemsmentioning

confidence: 99%

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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Section: The Spectral Extremal Graph Problemsmentioning

confidence: 99%

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

Li¹,

Peng

2022

Electron. J. Combin.

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“…On the one hand, various extensions and generalizations on inequality (4) in Nosal's theorem have been obtained in the literature; see, e.g., [51,38,19,25] for extension on K r+1 -free graphs with given order; see [7,40] for relations between cliques and spectral radius and [41,10,27] for surveys. Very recently, Lin, Ning and Wu [31,Theorem 1.4] proved a generalization on (4) for non-bipartite triangle-free graphs and provided a spectral version of Erdős' Theorem 1.2; see [29] for an alternative proof and refinement of spectral Turán theorem, and [28] for more stability theorems on spectral graph problems. In addition, Lin and Guo [32] proved an extension of nonbipartite graphs without short odd cycles.…”

Section: The Spectral Extremal Graph Problemsmentioning

confidence: 99%

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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“…; see [34,27] for a recent extension on graphs without short odd cycles, and [30] for more stability theorems on spectral graph problems.…”

mentioning

confidence: 99%

Refinement on spectral Turán's theorem

Li¹,

Peng²

2022

Preprint

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A well-known result in extremal spectral graph theory, known as Nosal's theorem, states that if G is a triangle-free graph on n vertices, then λ Algebra Appl. 427 (2007)] extended Nosal's theorem to K r+1 -free graphs for every integer r ≥ 2. This is known as the spectral Turán theorem. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] proved a refinement on Nosal's theorem for non-bipartite triangle-free graphs. In this paper, we provide alternative proofs for the result of Nikiforov and the result of Lin, Ning and Wu. Our proof can allow us to extend the later result to non-r-partite K r+1 -free graphs. Our result refines the theorem of Nikiforov and it also can be viewed as a spectral version of a theorem of Brouwer.

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New proofs of stability theorems on spectral graph problems

Cited by 3 publications

References 36 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

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

Refinement on spectral Turán's theorem

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