Malaysian Journal of Mathematical Sciences, June 2023, Vol. 17, No. 2
On the Non-Zero Divisor Graphs of Some Finite Commutative Rings
Zai, N. A. F. O., Sarmin, N. H., Khasraw, S. M. S., Gambo, I., and Zaid, N.
Corresponding Email: nhs@utm.my
Received date: 1 June 2020
Accepted date: 7 April 2023
Abstract:
The study of rings and graphs has been explored extensively by researchers. To gain a more effective understanding on the concepts of the rings and graphs, more researches on graphs of different types of rings are required. This manuscript provides a different study on the concepts of commutative rings and undirected graphs. The non-zero divisor graph, $\Gamma(R)$ of a ring $R$ is a simple undirected graph in which its set of vertices consists of all non-zero elements of $R$ and two different vertices are joint by an edge if their product is not equal to zero. In this paper, the commutative rings are the ring of integers modulo $n$ where $n=8k$ and $k\leq3$. The zero divisors are found first using the definition and then the non-zero divisor graphs are constructed. The manuscript explores some properties of non-zero divisor graph such as the chromatic number and the clique number. The result has shown that $\Gamma(\mathbb{Z}_{8k})$ is perfect.
Keywords: commutative rings; non-zero divisor graphs; ring; zero divisors
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