On normalized Laplacian spectrum of zero divisor graphs of commutative ring ℤn

S. Pirzada, Bilal A. Rather, T. A. Chishti, U. Samee


For a finite commutative ring ℤn with identity 1 ≠ 0, the zero divisor graph Γ(ℤn) 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 normalized Laplacian spectrum of the zero divisor graphs Γ(ℤn) for various values of n and characterize n for which Γ(ℤn) is normalized Laplacian integral. We also obtain bounds for the sum of graph invariant Sβ*(G)-the sum of the β-th power of the non-zero normalized Laplacian eigenvalues of Γ(ℤn).


normalized Laplacian matrix, normalized Laplacian spectrum, zero divisor graph

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DOI: http://dx.doi.org/10.5614/ejgta.2021.9.2.7


D. F. Anderson and P. S. Livingston, The zero divisor graph of a commutative ring, J. Algebra 217 (1999) 434–447.

I. Beck, Coloring of commutative rings, J. Algebra 116 (1988) 208–226.

S. Bozkurt and D. Bozkurt, On the sum of powers of normalized Laplacian eigenvalues of graphs, MATCH Commun. Math. Comput. Chem. 68 (2012) 917–930.

D.M. Cardoso, M.A. De Freitas, E. N. Martins and M Robbiano, Spectra of graphs obtained by a generalization of the join of graph operations, Discrete Math. 313 (2013) 733–741.

M. Cavers, The normalized Laplacian matrix and general Randic index of graphs, Thesis, University of Regina 2010.

S. Chattopadhyay, K. L. Patra and B. K. Sahoo, Laplacian eigenvalues of the zero divisor graph of the ring Zn, Linear Algebra Appl. 584 (2020) 267–286.

F. R. K. Chung, Spectral Graph Theory American Mathematical Society, Providence, 1997.

D. M. Cvetkovic, P. Rowlison, and S. Simic, An Introduction to Theory of Graph spectra Lon. Math. Society Student Text, 75. Cambridge University Press, Inc. UK 2010.

H. Chen and F. Zhang, Resistance distance and the normalized Laplacian spectrum, Discrete Appl. Math. 155 (2007) 654–661.

K.C. Das, A.D. Gungor, and S. Bozkurt, On the normalized Laplacian eigenvalues of graphs, Ars Combin. 118 (2015) 143–154.

R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, 1985.

S. Pirzada, An Introduction to Graph Theory, Universities Press, Orient BlackSwan, Hyderabad (2012).

S. Pirzada, B.A. Rather, A. Ahmad, and T.A. Chishti, On distance signless Laplacian spectrum of graphs and spectrum of zero divisor graphs of Zn, Linear Multilinear Algebra,


S. Pirzada, B.A. Wani, and A. Somasundaram, On the eigenvalues of zero divisor graph associated to finite commutative ring Zp^Mq^N, AKCE International J. Graphs Combinatorics,


S. Pirzada, B.A. Rather, and T.A. Chishti, On distance Laplacian spectrum of zero divisor graphs of Zn, Carpathian Mathematical Publications 13 (1) (2021), 48–57.


S. Pirzada, B.A. Rather, R.U. Shaban and Merajuddin, On signless Laplacian spectrum of the zero divisor graph of the ring Zn, Korean J. Mathematics 29 (1) (2021), 13–24.


B.A. Rather, S. Pirzada, T.A. Naikoo and Y. Shang, On Laplacian eigenvalues of the zerodivisor graph associated to the ring of integers modulo n, Mathematics, 9 (5) (2021), 482.


B.F. Wu, Y.Y. Lou and C.X. He, Signless Laplacian and normalized Laplacian on the H-join operation of graphs, Discrete Math. Algorithm. Appl. 6 (2014) (13 pages).

M. Young, Adjacency matrices of zero divisor graphs of integer modulo n, Involve 8 (2015), 753–761.

P. Zumstein, Comparison of Spectral Methods Through the Adjacency Matrix and the Laplacian of a Graph, Diplomat thesis, ETH Zurich, 2006


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