Integer divisor connectivity graph

M. Jorf; L. Oukhtite

doi:10.3842/umzh.v76i11.7863

M. Jorf Department of Mathematics, Faculty of Science and Technology, Sidi Mohamed Ben Abdellah University, Fez, Morocco
L. Oukhtite Department of Mathematics, Faculty of Science and Technology, Sidi Mohamed Ben Abdellah University, Fez, Morocco

DOI: https://doi.org/10.3842/umzh.v76i11.7863

Keywords: divisor, graph, prime number.

Abstract

UDC 512.5

Let $n$ be a nonprime integer. We introduce a new simple undirected graph and denote it by $MD(n),$ where the vertices are the proper divisors of $n$ and two vertices $x$ and $y$ are adjacent if $xy$ divides $n.$ We explore the connectedness of $MD(n)$ and provide detailed calculations for the degree of each vertex. In addition, we focus on the special case where $n = p^{\alpha},$ where $p$ is a prime positive integer and $\alpha\geq 3$ is a positive integer. For these instances, we explicitly determine the chromatic number $\chi$ and the clique number $\omega$ of $MD(n).$ Finally, we conclude that $\chi(MD(n)) = \omega(MD(n)).$

References

D. F. Anderson, A. Badawi, On the zero-divisor graph of a ring, Comm. Algebra, 36, № 8, 3073–3092 (2008). DOI: https://doi.org/10.1080/00927870802110888

D. F. Anderson, A. Badawi, The total graph of a commutative ring, J. Algebra, 320, № 7, 2706–2719 (2008). DOI: https://doi.org/10.1016/j.jalgebra.2008.06.028

D. F. Anderson, P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217, 434–447 (1999). DOI: https://doi.org/10.1006/jabr.1998.7840

I. Beck, Coloring of commutative rings, J. Algebra, 116, № 1, 208–226 (1988). DOI: https://doi.org/10.1016/0021-8693(88)90202-5

N. Biggs, Algebraic graph theory (Cambridge Mathematical Library), 2nd ed., Cambridge Univ. Press, Cambridge (1993).

T. T. Chelvam, T. Asir, On the genus of the total graph of a commutative ring, Comm. Algebra, 41, № 1, 142–153 (2013). DOI: https://doi.org/10.1080/00927872.2011.624147

T. Fenstermacher, E. Gegner, Zero-divisor graph of $2 × 2$ upper triangular matrix rings over $Z,_n$, Missouri J. Math. Sci., 26, № 2, 151–167 (2014). DOI: https://doi.org/10.35834/mjms/1418931956

H. Kumar, K. L. Patra, B. K. Sahoo, Proper divisor graph of a positive integer, Integers, 21, Article A65 (2021).

V. I. Voloshin, Introduction to graph theory, Nova Sci. Publ. Inc, New York (2009).

D. B. West, Introduction to graph theory, second ed., Prentice Hall of India Private Limited, New Delhi (2003).