Adjacency eigenvalues of graphs without short odd cycles

Li, Shuchao; Sun, Wanting; Yu, Yuantian

doi:10.1016/j.disc.2021.112633

Cited by 23 publications

(14 citation statements)

References 39 publications

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“…In addition, Lin and Guo [32] proved an extension of nonbipartite graphs without short odd cycles. This result was also independently proved by Li, Sun and Yu [26,Theorem 1.6] using a different method.…”

Section: The Spectral Extremal Graph Problemssupporting

confidence: 64%

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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 Problemssupporting

confidence: 64%

“…Theorem 1.4 confirmed the case r = 2; see [42,26] for recent progress. This conjecture of Bollobás and Nikiforov remains open for the case r ≥ 3.…”

Section: The Spectral Extremal Graph Problemsmentioning

confidence: 64%

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

Li¹,

Peng²

2022

Preprint

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“…Theorem 4 confirmed the case r = 2; see [42,26] for recent progress. This conjecture of Bollobás and Nikiforov remains open for the case r 3.…”

Section: The Spectral Extremal Graph Problemsmentioning

confidence: 54%

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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“…More precisely, what is the maximum spectral radius among all n-vertex H-free graphs? In the past few decades, the problem has been investigated by many researchers for various graphs H, such as, the complete graphs [23,34,20], the complete bipartite graphs [2,24], the books and theta graphs [38], the friendship graphs [5,37,40], the intersecting odd cycles [18,11], the intersecting cliques [12], the paths and linear forests [25,10], the odd wheels [6], the quadrilateral [23,35], the hexagon [36], the short odd cycles [15,17,19], the square of a path [41], the fan graph [32]. We refer the reader to [27] for a comprehensive survey.…”

Section: Spectral Extremal Graphs For Friendship Graphsmentioning

confidence: 99%

Spectral extremal graphs without intersecting triangles as a minor

He¹,

Li²,

Feng³

2023

Preprint

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Let F s be the friendship graph obtained from s triangles by sharing a common vertex. For fixed s ≥ 2 and sufficiently large n, the F s -free graphs of order n which attain the maximal spectral radius was firstly characterized by Cioabȃ, Feng, Tait and Zhang [Electron. J. Combin. 27 (4) (2020)], and later uniquely determined by Zhai, Liu and Xue [Electron. J. Combin. 29 (3) ( 2022)]. Recently, the spectral extremal problems was widely studied for graphs containing no H as a minor. For instance, Tait [J. Combin. Theory Ser. A 166 (2019)], Zhai and Lin [J. Combin. Theory Ser. B 157 (2022)] solved the case H = K r and H = K s,t , respectively. Motivated by these results, we consider the spectral extremal problems in the case H = F s . We shall prove that K s ∨ I n−s is the unique graph that attain the maximal spectral radius over all n-vertex F s -minor-free graphs. Moreover, let Q t be the graph obtained from t copies of the cycle of length 4 by sharing a common vertex. We also determine the unique Q t -minor-free graph attaining the maximal spectral radius. Namely, K t ∨ M n−t , where M n−t is a graph obtained from an independent set of order n − t by embedding a matching consisting of ⌊ n−t 2 ⌋ edges.

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Adjacency eigenvalues of graphs without short odd cycles

Cited by 23 publications

References 39 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

Spectral extremal graphs without intersecting triangles as a minor

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