Signless Laplacian spectrum of power graphs of finite cyclic groups

Banerjee, Subarsha; Adhikari, Avishek

doi:10.1016/j.akcej.2019.03.009

Cited by 17 publications

(7 citation statements)

References 10 publications

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“…In [9], it is shown that Laplacian spectral radius of power graph of any finite group coincides with the order of group G. Spectral properties of adjacency matrix of P(G) were investigated in [15]. Other spectral results of power graphs can be seen in [5,18,19,20,21].…”

Section: Accepted Manuscriptmentioning

confidence: 99%

On distance signless Laplacian spectra of power graphs of the integer modulo group

Rather

Pirzada

Naikoo

2022

Art Discrete Appl. Math.

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For a finite group G, the power graph P(G) is a simple connected graph whose vertex set is the set of elements of G and two distinct vertices are adjacent if and only if one is a power of the other. In this article, we obtain the distance signless Laplacian spectrum of power graphs of the integer modulo groups Z n . We characterize the values of n, for which power graphs of Z n is distance signless Laplacian integral.

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Section: Accepted Manuscriptmentioning

confidence: 99%

On distance signless Laplacian spectra of power graphs of the integer modulo group

Rather

Pirzada

Naikoo

2022

Art Discrete Appl. Math.

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“…The adjacency spectrum, the Laplacian, the normalized Laplacian and the signless Laplacian spectrum of power graphs of finite cyclic and dihedral groups have been investigated in [13,16,28,29,43,35,44].…”

Section: Normalized Distance Laplacian Eigenvalues Of Power Graphs Of...mentioning

confidence: 99%

On normalized distance Laplacian eigenvalues of graphs and applications to graphs defined on groups and rings

RATHER¹,

GANIE²,

AOUCHICHE³

2022

CJM

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The normalized distance Laplacian matrix of a connected graph $ G $, denoted by $ D^{\mathcal{L}}(G) $, is defined by $ D^{\mathcal{L}}(G)=Tr(G)^{-1/2}D^L(G)Tr(G)^{-1/2}, $ where $ D(G) $ is the distance matrix, the $D^{L}(G)$ is the distance Laplacian matrix and $ Tr(G)$ is the diagonal matrix of vertex transmissions of $ G. $ The set of all eigenvalues of $ D^{\mathcal{L}}(G) $ including their multiplicities is the normalized distance Laplacian spectrum or $ D^{\mathcal{L}} $-spectrum of $G$. In this paper, we find the $ D^{\mathcal{L}} $-spectrum of the joined union of regular graphs in terms of the adjacency spectrum and the spectrum of an auxiliary matrix. As applications, we determine the $ D^{\mathcal{L}} $-spectrum of the graphs associated with algebraic structures. In particular, we find the $ D^{\mathcal{L}} $-spectrum of the power graphs of groups, the $ D^{\mathcal{L}} $-spectrum of the commuting graphs of non-abelian groups and the $ D^{\mathcal{L}} $-spectrum of the zero-divisor graphs of commutative rings. Several open problems are given for further work.

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“…Let G be a finite group of order n with identity element e. Chakrabarty et al [7] defined the undirected power graph P(G) of a group G as an undirected graph with vertex set as G and two vertices x, y ∈ G are adjacent if and only if one is the positive power of other, that is, x i = y or y j = x, for positive integers i, j with 2 ≤ i, j ≤ n. For some recent work on power graphs, we refer to [1,5,7,8,15] and the references therein. The adjacency spectrum, the Laplacian and the signless Laplacian spectrum of power graphs of finite cyclic and dihedral groups has been investigated in [3,6,11,16,20]. (n i + 1), so the order of the graph…”

Section: A α -Spectrum Of Power Graphs Of Certain Finite Groupsmentioning

confidence: 99%

On $A_α$-spectrum of joined union of graphs and its applications to power graphs of finite groups

Rather¹,

Ganie²,

Pirzada³

2021

Preprint

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For a simple graph G, the generalized adjacency matrix A α (G) is defined aswhere A(G) is the adjacency matrix and D(G) is the diagonal matrix of vertex degrees of G. This matrix generalises the spectral theories of the adjacency matrix and the signless Laplacian matrix of G. In this paper, we find A α -spectrum of the joined union of graphs in terms of spectrum of adjacency matrices of its components and the zeros of the characteristic polynomials of an auxiliary matrix determined by the joined union. We determine the A α -spectrum of join of two regular graphs, the join of a regular graph with the union of two regular graphs of distinct degrees. As an applications, we investigate the A α -spectrum of certain power graphs of finite groups.

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Signless Laplacian spectrum of power graphs of finite cyclic groups

Cited by 17 publications

References 10 publications

On distance signless Laplacian spectra of power graphs of the integer modulo group

On distance signless Laplacian spectra of power graphs of the integer modulo group

On normalized distance Laplacian eigenvalues of graphs and applications to graphs defined on groups and rings

On $A_α$-spectrum of joined union of graphs and its applications to power graphs of finite groups

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