Eigenvalue of Adjacent Matrix of Zero Divisor Graphs on Rings
top of page
Asian Institute of Research, Journal Publication, Journal Academics, Education Journal, Asian Institute
Asian Institute of Research, Journal Publication, Journal Academics, Education Journal, Asian Institute

Engineering and Technology Quarterly Reviews

ISSN 2622-9374

Screen Shot 2018-08-15 at 7.28.21 PM.png
Screen Shot 2018-08-15 at 7.28.06 PM.png
Screen Shot 2018-08-15 at 7.28.12 PM.png
Screen Shot 2018-08-15 at 7.28.27 PM.png
crossref
doi
open access

Published: 03 September 2020

Eigenvalue of Adjacent Matrix of Zero Divisor Graphs on Rings

Hemati Sherin

Bamyan University, Afghanistan

journal of social and political sciences
pdf download

Download Full-Text Pdf

doi

10.5281/zenodo.4011991

Pages: 63-66

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

Abstract

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)).

References

  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.

bottom of page