Spectral Extremal Results with Forbidding Linear Forests

Chen, Mingzhu; Liu, A-Ming; Zhang, Xiao‐Dong

doi:10.1007/s00373-018-1996-3

Cited by 36 publications

(17 citation statements)

References 19 publications

(21 reference statements)

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“…For linear forest, Chen, Liu, and Zhang [5] in 2019 determined the maximum spectral radius of an F ℓ -free graph of order n and characterized all the extremal graphs, which is the spectral counterpart of Theorems 1.4 also.…”

Section: The Spectral Turán Extremal Problemmentioning

confidence: 99%

The bipartite Turan number and spectral extremum for linear forests

Chen¹,

Wang²,

Yuan³

et al. 2022

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The bipartite Turán number of a graph H, denoted by ex(m, n; H), is the maximum number of edges in any bipartite graph G = (X, Y ; E) with |X| = m and |Y | = n which does not contain H as a subgraph. In this paper, we determined ex(m, n; F ℓ ) for arbitrary ℓ and appropriately large n with comparing to m and ℓ, where F ℓ is a linear forest which consists of ℓ vertex disjoint paths. Moreover, the extremal graphs have been characterized. Furthermore, these results are used to obtain the maximum spectral radius of bipartite graphs which does not contain F ℓ as a subgraph and characterize all extremal graphs which attain the maximum spectral radius.

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Section: The Spectral Turán Extremal Problemmentioning

confidence: 99%

The bipartite Turan number and spectral extremum for linear forests

Chen¹,

Wang²,

Yuan³

et al. 2022

Preprint

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“…In the past decade, this problem has been studied for various choices of H, such as for complete graphs [13], complete bipartite graphs [1,11], and for cycles or paths of specified length [5,9,10,14,15]. More results on spectral extremal problems can be found in [2,3,4,12].…”

Section: Introductionmentioning

confidence: 99%

Spectral radius conditions for the existence of all subtrees of diameter at most four

Liu¹,

Broersma

Wang³

2021

Preprint

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Let µ(G) denote the spectral radius of a graph G. We partly confirm a conjecture due to Nikiforov, which is a spectral radius analogue of the well-known Erdős-Sós Conjecture that any tree of order t is contained in a graph of average degree greater than t − 2. Let S n,k = K k ∨ K n−k , and let S + n,k be the graph obtained from S n,k by adding a single edge joining two vertices of the independent set of S n,k . In 2010, Nikiforov conjectured that for a given integer k, every graph G of sufficiently large order n with µ(G) ≥ µ(S + n,k ) contains all trees of order 2k + 3, unless G = S + n,k . We confirm this conjecture for trees with diameter at most four, with one exception. In fact, we prove the following stronger result for k ≥ 8. If a graph G with sufficiently large order n satisfies µ(G) ≥ µ(S n,k ) and G = S n,k , then G contains all trees of order 2k + 3 with diameter at most four, except for the tree obtained from a star K 1,k+1 by subdividing each of its k + 1 edges once.

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“…The Brualdi-Solheid-Turán type problem is how to determine the maximum spectral radius of an H-free graph of order n and characterize those graphs which attain the maximum spectral radius. The Brualdi-Solheid-Turán type problem of the adjacent spectral spectral radius has been studied for various kinds of H such as the complete graph [16], the complete bipartite graph [18], the cycles or paths of specified length [17], the linear forest [3], and star forest [4] and so on. In addition, the Brualdi-Solheid-Turán type problem of the signless Laplacian spectral radius has also been investigated extensively in the literature.…”

Section: Introductionmentioning

confidence: 99%

The signless Laplacian spectral radius of graphs without intersecting odd cycles

Chen¹,

Liu²,

Zhang³

2021

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Let Fa 1 ,...,a k be a graph consisting of k cycles of odd length 2a1 + 1, . . . , 2a k + 1, respectively which intersect in exactly a common vertex, where k ≥ 1 and a1 ≥ a2 ≥ • • • ≥ a k ≥ 1. In this paper, we present a sharp upper bound for the signless Laplacian spectral radius of all Fa 1 ,...,a k -free graphs and characterize all extremal graphs which attain the bound. The stability methods and structure of graphs associated with the eigenvalue are adapted for the proof.

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Spectral Extremal Results with Forbidding Linear Forests

Cited by 36 publications

References 19 publications

The bipartite Turan number and spectral extremum for linear forests

The bipartite Turan number and spectral extremum for linear forests

Spectral radius conditions for the existence of all subtrees of diameter at most four

The signless Laplacian spectral radius of graphs without intersecting odd cycles

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