Engineering and Technology Quarterly Reviews

ISSN 2622-9374

Published: 03 September 2020

Eigenvalue of Adjacent Matrix of Zero Divisor Graphs on Rings

Hemati Sherin

Bamyan University, Afghanistan

pdf download

Download Full-Text Pdf


Pages: 63-66

Keywords: Eigenvalue, Adjacency Matrix, Zero Devisors Graph of Commutative Ring


Let R be a commutative ring with identity 1≠0 and T be the ring of all nxn upper triangular matrices over R. The zero-devisor graph of T denoted by T(Tn(R)). In this paper, I define the adjacent Matrix of T(R) and T(Tn(R)). Then I describe the relation between the non-zero Eigenvalues of adjacent Matrix of these graph and edges. After I use these result to determination of Eigenvalue adjacent matrix of T(T2(R)).


  1. Anderson, David F, and Philip S Livingston. 1999. "The zero-divisor graph of a commutative ring." Journal of algebra 217 (2):434-447.

  2. Anderson, DD, and M Naseer. 1993. "Beck′ s coloring of a commutative ring." Journal of algebra 159 (2):500-514.

  3. Anderson, David F. 2008. "On the diameter and girth of a zero-divisor graph, II." Houston journal of mathematics 34 (2):361-372.

  4. Bapat, Ravindra B. 2010. Graphs and matrices. Vol. 27: Springer.

  5. Beck, Istvan. 1988. "Coloring of commutative rings." Journal of algebra 116 (1):208-226.

  6. Harary, Frank. 1962. "The determinant of the adjacency matrix of a graph." Siam Review 4 (3):202-210.

  7. Jørgensen, Leif K. 2005. "Rank of adjacency matrices of directed (strongly) regular graphs." Linear algebra and its applications 407:233-241.

  8. Li, Aihua, and Ralph P Tucci. 2013. "Zero divisor graphs of upper triangular matrix rings." Communications in Algebra 41 (12):4622-4636.

  9. Li, B. 2011. "Zero-divisor graph of triangular matrix rings over commutative rings." Int. J. Algebra 5:255-260.

  10. Redmond, Shane P. 2002. "The zero-divisor graph of a non-commutative ring." Internat. J. Commutative Rings. v1 i4:203-211.

About Us

The Asian Institute of Research is an online and open-access platform to publish recent research and articles of scholars worldwide. Founded in 2018 and based in Indonesia, the Institute serves as a platform for academics, educators, scholars, and students from Asia and around the world, to connect with one another. The Institute disseminates research that is proven or predicted to be of significant influence for the general public.

Contact Us

Please send all inquiries to the email:

Business Address:

5th Floor, Kavling 507, Fajar Graha Pena Tower, Jl. Urip Sumohardjo No.20, Makassar, Indonesia 90234

Copyright © 2018 The Asian Institute of Research. All rights reserved

Stay Connected

  • Instagram - Black Circle
  • Facebook - Black Circle
  • LinkedIn - Black Circle