On a conjecture for the signless Laplacian spectral radius of cacti with given matching number

Shen, Yun; You, Lihua; Zhang, Minjie; Li, Shuchao

doi:10.1080/03081087.2016.1189494

Cited by 13 publications

(5 citation statements)

References 43 publications

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“…Chen and Zhou [5] investigated some upper bounds of the signless Laplacian spectral radius of cactus graphs. The signless Laplacian spectral radius of cacti with given matching number are obtained by Shen et al [17]. Some results for spectral radius on cacti with k pendant vertices are studied Wu et al [18].…”

Section: Introductionmentioning

confidence: 97%

Sharp upper bounds of $A_\alpha$-spectral radius of cacti with given pendant vertices

Wang

Liu

2021

Hacettepe Journal of Mathematics and Statistics

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For α ∈ [0, 1], let A α (G) = αD(G)+(1−α)A(G) be A α -matrix, where A(G) is the adjacent matrix and D(G) is the diagonal matrix of the degrees of a graph G. Clearly, A 0 (G) is the adjacent matrix and 2A 1 2 is the signless Laplacian matrix. A connected graph is a cactus graph if any two cycles of G have at most one common vertex. We first propose the result for subdivision graphs, and determine the cacti maximizing A α -spectral radius subject to fixed pendant vertices. In addition, the corresponding extremal graphs are provided. As consequences, we determine the graph with the A α -spectral radius among all the cacti with n vertices; we also characterize the n-vertex cacti with a perfect matching having the largest A α -spectral radius.

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Section: Introductionmentioning

confidence: 97%

Sharp upper bounds of $A_\alpha$-spectral radius of cacti with given pendant vertices

Wang

Liu

2021

Hacettepe Journal of Mathematics and Statistics

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“…Let G be an n-vertex graph with vertex set V (G) = {v 1 , v 2 , • • • , v n } and edge set E(G), then the signless Laplacian matrix of G is defined as Q(G) = D(G) + A(G), where D(G) is a diagonal matrix of vertex degrees and A(G) is the adjacency matrix. Since its introduction, there have existed lots of investigations on the signless Laplacian spectrum (see [7,21] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Signless Laplacian spectral radius and matching in graphs

Liu¹,

Pan²,

Li³

2020

Preprint

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The signless Laplacian matrix of a graph G is given by Q(G) = D(G) + A(G), where D(G) is a diagonal matrix of vertex degrees and A(G) is the adjacency matrix. The largest eigenvalue of Q(G) is called the signless Laplacian spectral radius, denoted by q 1 = q 1 (G). In this paper, some properties between the signless Laplacian spectral radius and perfect matching in graphs are establish. Let r(n) be the largest root of equation x 3 − (3n − 7)x 2 + n(2n − 7)x − 2(n 2 − 7n + 12) = 0. We show that G has a perfect matching for n = 4 or n ≥ 10, if q 1 (G) > r(n), and for n = 6 or n = 8, if q 1 (G) > 4 + 2 √ 3 or q 1 (G) > 6 + 2 √ 6 respectively, where n is a positive even integer number. Moreover, there exists graphs√ 3 and a graph K 3 ∨ K 5 such that q 1 (K 3 ∨ K 5 ) = 6 + 2 √ 6. These graphs all have no prefect matching.

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“…Wu et al [16] found the spectral radius of cacti with k-pendant vertices. Shen et al [17] studied the signless Laplacian spectral radius of cacti with given matching number.…”

Section: Introductionmentioning

confidence: 99%

On the Aα-Spectral Radii of Cactus Graphs

Wang

Liu

et al. 2020

Mathematics

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Let A ( G ) be the adjacent matrix and D ( G ) the diagonal matrix of the degrees of a graph G, respectively. For 0 ≤ α ≤ 1 , the A α -matrix is the general adjacency and signless Laplacian spectral matrix having the form of A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . Clearly, A 0 ( G ) is the adjacent matrix and 2 A 1 2 is the signless Laplacian matrix. A cactus is a connected graph such that any two of its cycles have at most one common vertex, that is an extension of the tree. The A α -spectral radius of a cactus graph with n vertices and k cycles is explored. The outcomes obtained in this paper can imply some previous bounds from trees to cacti. In addition, the corresponding extremal graphs are determined. Furthermore, we proposed all eigenvalues of such extremal cacti. Our results extended and enriched previous known results.

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On a conjecture for the signless Laplacian spectral radius of cacti with given matching number

Cited by 13 publications

References 43 publications

Sharp upper bounds of $A_\alpha$-spectral radius of cacti with given pendant vertices

Sharp upper bounds of $A_\alpha$-spectral radius of cacti with given pendant vertices

Signless Laplacian spectral radius and matching in graphs

On the Aα-Spectral Radii of Cactus Graphs

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