Research Journal of Science and Technology

Open Access Open Access  Restricted Access Subscription Access
Open Access Open Access Open Access  Restricted Access Restricted Access Subscription Access

On the Complement of the Intersection Graph of Zero-Divisors of the Ring Zn


Affiliations
1 Department of Mathematics, S.V. University, Tirupati, A.P., -517502, India
     

   Subscribe/Renew Journal


For the ring of integers modulo 𝑛, we study the complement of the intersection graph of zero-divisors is denoted by 𝐺𝑍′ (𝑍𝑛)Μ… and is defined as a simple undirected graph whose vertices are the set of all nonzero zero-divisors of the ring 𝑍𝑛 and in which two distinct vertices are joined by an edge if and only if their corresponding principal ideals have zero intersection. We determine the necessary and sufficient condition for adjacency of vertices in the graph 𝐺𝑍′ (𝑍𝑛)Μ…. Also, we investigate the connectedness and further calculate the radius and diameter of the graph 𝐺𝑍′ (𝑍𝑛)Μ… for all characterizations of 𝑛.

Keywords

Intersection Graph, Zero-Divisors, Principal Ideal, Connected Graph, Eccentricity, Radius, Diameter.
Subscription Login to verify subscription
User
Notifications
Font Size


  • Anderson David F. and Livingston Philip S., The Zero-Divisor Graph of a Commutative Ring, J. of Algebra, 1999, 217, 434-447.

  • Apostol Tom M., Introduction to Analytical Number Theory, Springer International Student First Edition, Narosa publishing house,1989.

  • Bondy J.A. and Murty U.S.R., Graph Theory with Applications, Macmillan Press Ltd, Great Britain, 1976.

  • Bosak J., The graphs of semigroups, Theory of graphs and its applications, Proc. Sympos. Smolenice (June1963), Academic Press, New York, 1965, 119-125.

  • Chakrabarty Ivy, Shamik Ghosh, Mukherjee T.K., Sen M.K., Intersection Graphs of Ideals of Rings, Discrete Mathematics, 2009, 309, 5381-5392.

  • Godsil C. and Royle G., Algebraic Graph Theory, Graduate text in mathematics, Springer-Verlag, New York, 2001.

  • Kaplansky I., Commutative rings, rev. ed., Univ. Of Chicago press, Chicago, 1974.

  • Rosen K.H., Elementary number theory and its applications, Addison-Wesley, 1984.

  • Sajana Shaik, Srimitra K.K., Bharathi D., β€œIntersection Graph of Zero-divisors of a Finite Commutative Ring”, International Journal of Pure and Applied Mathematics, 2016, Volume 109, No. 7, 51-58.

  • Sajana Shaik, Bharathi D., Srimitra K.K., On the Reduced Intersection Graph of a Ring 𝑍𝑛, International Journal of Advanced Research in Computer Science, 2017, Vol. 8, No. 6, July (Special Issue III), 200-202.

  • West Douglas B., Introduction to Graph Theory, Prentice-Hall of India Private Limited, New Delhi, 2003.

  • Zelinka Bohdan, Liberec, Intersection Graphs of Finite Abelian Groups, Czechoslovak Mathematical journal, 1975, V.25, No 2, 171-174.


Abstract Views: 156

PDF Views: 0




  • On the Complement of the Intersection Graph of Zero-Divisors of the Ring Zn

Abstract Views: 156  |  PDF Views: 0

Authors

Shaik Sajana
Department of Mathematics, S.V. University, Tirupati, A.P., -517502, India
D. Bharathi
Department of Mathematics, S.V. University, Tirupati, A.P., -517502, India
K. K. Srimitra
Department of Mathematics, S.V. University, Tirupati, A.P., -517502, India

Abstract


For the ring of integers modulo 𝑛, we study the complement of the intersection graph of zero-divisors is denoted by 𝐺𝑍′ (𝑍𝑛)Μ… and is defined as a simple undirected graph whose vertices are the set of all nonzero zero-divisors of the ring 𝑍𝑛 and in which two distinct vertices are joined by an edge if and only if their corresponding principal ideals have zero intersection. We determine the necessary and sufficient condition for adjacency of vertices in the graph 𝐺𝑍′ (𝑍𝑛)Μ…. Also, we investigate the connectedness and further calculate the radius and diameter of the graph 𝐺𝑍′ (𝑍𝑛)Μ… for all characterizations of 𝑛.

Keywords


Intersection Graph, Zero-Divisors, Principal Ideal, Connected Graph, Eccentricity, Radius, Diameter.

References