On the Signless Laplacian Spectral Radius of Cacti

In the past decades, graphs that are determined by their spectrum have received more attention, since they have been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. An important part of spectral graph theory is devoted to determining whether given graphs or classes of graphs are determined by their spectra or not. So, finding and introducing any class of graphs which are determined by their spectra can be an interesting and important problem. A graph is said to be DQS if there is no other non-isomorphic graph with the same signless Laplacian spectrum. For a DQS graph G, we show that G ∪ rK1 ∪ sK2 is DQS under certain conditions, where r, s are natural numbers and K1 and K2 denote the complete graphs on one vertex and two vertices, respectively. Applying these results, some DQS graphs with independent edges and isolated vertices are obtained.

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“…even unicyclic). Lemma 2.5 [16] Let H be a proper subgraph of a connected graph G. Then, q 1 (G) > q 1 (H).…”

Section: Lemma 23 [19] For Any Connected Bipartite Graphmentioning

confidence: 99%

“…with equality if and only if G is a non-bipartite bicyclic graph with C 4 as its induced subgraph. Lemma 2.5 [16] Let H be a proper subgraph of a connected graph G. Then, q 1 (G) > q 1 (H).…”

Section: Lemma 24 [35] For Any Graphmentioning

confidence: 99%

Graphs determined by signless Laplacian spectra

Abdian

Behmaram

Fath-Tabar

2020

AKCE International Journal of Graphs and Combinatorics

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“…Furthermore, we explore all eigenvalues of such extremal cacti. By using these outcomes, some previous results can be deduced, see [2,5,10].…”

Section: Introductionmentioning

confidence: 95%

“…Borovićanin and Petrović investigate the properties of cacti with n vertices [2]. Chen and Zhou [5] obtain the upper bound of the signless Laplacian spectral radius of cacti. Wu et al [14] find the spectral radius of cacti with k-pendant vertices.…”

Section: Introductionmentioning

confidence: 99%

On the largest $A_α$-spectral radius of cacti

Wang,

Liu

et al. 2018

Preprint

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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 0is given by Nikiforov. Clearly, A 0 (G) is the adjacent matrix and 2A 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 previous bounds of Nikiforov et al., and Lovász and Pelikán. 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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“…For instance, the properties of cacti with n vertices [3] are explored by Borovićanin and Petrović. 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].…”

Section: Introductionmentioning

confidence: 99%

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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On the Signless Laplacian Spectral Radius of Cacti

Cited by 5 publications

References 15 publications

Graphs determined by signless Laplacian spectra

Graphs determined by signless Laplacian spectra

On the largest $A_α$-spectral radius of cacti

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

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