Korean J. Math.  Vol 29, No 1 (2021)  pp.13-24
DOI: https://doi.org/10.11568/kjm.2021.29.1.13

On signless Laplacian spectrum of the zero divisor graphs of the ring $\mathbb{Z}_{n}$

Shariefuddin Pirzada, Bilal Rather, Rezwan Ul Shaban, S Merajuddin

Abstract


For a finite commutative ring $ R $ with identity $ 1\neq 0 $, the zero divisor graph $ \Gamma(R) $ is a simple connected graph having vertex set as the set of nonzero zero divisors of $ R $, where two vertices $ x $ and $ y $ are adjacent if and only if $ xy=0$. We find the signless Laplacian spectrum of the zero divisor graphs $ \Gamma(\mathbb{Z}_{n}) $ for various values of $ n$. Also, we find signless Laplacian spectrum of $ \Gamma(\mathbb{Z}_{n}) $ for $ n=p^z , z\geq 2 $, in terms of signless Laplacian spectrum of its components and zeros of the characteristic polynomial of an auxiliary matrix. Further, we characterise $ n $ for which zero divisor graph $  \Gamma(\mathbb{Z}_{n})  $  are signless Laplacian integral.


Keywords


Signless Laplacian matrix; zero divisor graph, finite commutative ring

Subject classification

05C50, 05C12, 15A18

Sponsor(s)

SERB DST

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