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

Rather, Bilal A.; Ganie, Hilal A.; Pirzada, S.

doi:10.1142/s0219498823502572

Cited by 5 publications

(2 citation statements)

References 17 publications

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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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