Ordering trees by their Laplacian spectral radii

Yu, Aimei; Lu, M.; Tian, Feng

doi:10.1016/j.laa.2005.02.043

Cited by 13 publications

(4 citation statements)

References 7 publications

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Order By: Relevance

“…Many approaches to ordering trees by their spectra are contained in the literature (see [5,7,12,10]). Let T be a tree of order n. It is well known that (T ) 1, and (T ) = 1 if and only if T = K 1,n−1 .…”

Section: Introductionmentioning

confidence: 99%

The six classes of trees with the largest algebraic connectivity

Yuan

Shao

Zhang

2008

Discrete Applied Mathematics

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In this paper, we study the algebraic connectivity (T ) of a tree T. We introduce six Classes (C1)-(C6) of trees of order n, and prove that if T is a tree of order n 15, then (T ) 2 − √ 3 if and only if T ∈ 6 i=1 Ci, where the equality holds if and only if T is a tree in the Class (C6). At the same time we give a complete ordering of the trees in these six classes by their algebraic connectivity. In particular, we show that (T i ) > (T j ) if 1 i < j 6 and T i is any tree in the Class (Ci) and T j is any tree in the Class (Cj ). We also give the values of the algebraic connectivity of the trees in these six classes. As a technique used in the proofs of the above mentioned results, we also give a complete characterization of the equality case of a well-known relation between the algebraic connectivity of a tree T and the Perron value of the bottleneck matrix of a Perron branch of T.

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Section: Introductionmentioning

confidence: 99%

The six classes of trees with the largest algebraic connectivity

Yuan

Shao

Zhang

2008

Discrete Applied Mathematics

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“…Let G be a C 4 -free k-cyclic graph of order n. If k = 0 then G is a tree. In [6,15], the authors determined the first eight Laplacian spectral radius of trees of order n. For a bipartite graph G, by [3], we know that L(G) and Q(G) have the same eigenvalues. A tree is a bipartite graph, so the results that are obtained by [6,15] hold also for the signless Laplacian spectral radius of trees of order n. Therefore, in what follows, assume that k ≥ 1.…”

Section: Resultsmentioning

confidence: 99%

On the signless Laplacian spectral radius of $C_{4}$-free $k$-cyclic graphs

Kong,

Wang

2016

Preprint

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A k-cyclic graph is a connected graph of order n and size n + k − 1. In this paper, we determine the maximal signless Laplacian spectral radius and the corresponding extremal graph among all C 4 -free k-cyclic graphs of order n. Furthermore, we determine the first three unicyclic, and bicyclic, C 4 -free graphs whose spectral radius of the signless Laplacian is maximal. Similar results are obtained for the (combinatorial) Laplacian.

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“…The trees which take the first thirteen positions have been characterized [12,2,11,3]. Their results can be combined into the following Theorem 3.2.…”

Section: A Relation Between µ(T ) and ∆(T ) Of A Tree Tmentioning

confidence: 99%

On the Laplacian spectral radii of trees

Yuan

Shan

Liu

2009

Discrete Mathematics

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a b s t r a c tLet ∆(T ) and µ(T ) denote the maximum degree and the Laplacian spectral radius of a tree T , respectively. Let T n be the set of trees on n vertices, and T c n = {T ∈ T n | ∆(T ) = c}. In this paper, we determine the two trees which take the first two largest values of µ(T ) of the trees T in T c n when c ≥ n 2 . And among the trees in T c n , the tree which alone minimizes the Laplacian spectral radius is characterized. We also prove that for two trees T 1 and T 2 in T n (n ≥ 6), if ∆(T 1 ) > ∆(T 2 ) and ∆(T 1 ) ≥ n 2 + 1, then µ(T 1 ) > µ(T 2 ). As an application of these results, we give a general approach about extending the known ordering of trees in T n by their Laplacian spectral radii.

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Ordering trees by their Laplacian spectral radii

Cited by 13 publications

References 7 publications

The six classes of trees with the largest algebraic connectivity

The six classes of trees with the largest algebraic connectivity

On the signless Laplacian spectral radius of $C_{4}$-free $k$-cyclic graphs

On the Laplacian spectral radii of trees

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