On the non-zero divisor graphs of some finite commutative rings

• Main Authors: Omar Zai, Nur Athirah Farhana, Sarmin, Nor Haniza, Khasraw, Sanhan Muhammad Salih, Gambo, Ibrahim, Zaid, Nurhidayah
• Format: Article
• Language: English
• Published: Universiti Putra Malaysia 2023
• Subjects: QA Mathematics
• Online Access: http://eprints.utm.my/105396/1/NorHanizaSarmin2023_OntheNonZeroDivisorGraphsofSome.pdf; http://eprints.utm.my/105396/; http://dx.doi.org/10.47836/mjms.17.2.02
• Institution: Universiti Teknologi Malaysia

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, Γ(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 ≤ 3. 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 Γ(Z8k) is perfect.