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

Keywords: Laplacian matrix, distance Laplacian matrix, commutative ring, zero divisor graph
Published online: 2021-03-29

Abstract


For a finite commutative ring $\mathbb{Z}_{n}$ with identity $1\neq 0$, the zero divisor graph $\Gamma(\mathbb{Z}_{n})$ is a simple connected graph having vertex set as the set of non-zero zero divisors, where two vertices $x$ and $y$ are adjacent if and only if $xy=0$. We find the distance Laplacian spectrum of the zero divisor graphs $\Gamma(\mathbb{Z}_{n})$ for different values of $n$. Also, we obtain the distance Laplacian spectrum of $\Gamma(\mathbb{Z}_{n})$ for $n=p^z$, $z\geq 2$, in terms of the Laplacian spectrum. As a consequence, we determine those $n$ for which zero divisor graph $\Gamma(\mathbb{Z}_{n})$ is distance Laplacian integral.

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How to Cite
(1)
Pirzada S., Rather B., Chishti T. On Distance Laplacian Spectrum of Zero Divisor Graphs of the Ring $\mathbb{Z}_{n}$. Carpathian Math. Publ. 2021, 13 (1), 48-57.