On the Strong Annihilator Graph of a Commutative Ring

Subject Areas : Algebra

Nazi Abachi ¹ , Shervin Sahebi ²

1 - Department of Mathematics, Buinzahra Branch, Islamic Azad University, Buinzahra, Iran
2 - Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran

Received: 2022-04-10 Accepted : 2023-11-13 Published : 2024-09-26

Keywords: گراف تام, مقسوم علیه صفر, گراف کردال, گراف کامل,

Abstract :

Let be a commutative ring with identity, and let be the set of zero-divisors of . Anderson and Badawi introduced the well-known total graph of , denoted by , as the graph with the vertex set all elements of , and two distinct vertices and are adjacent if and only if . Also, they studied a subgraph of with the vertex set denoted by . In this total graph, two distinct vertices and are adjacent if and only if . In this paper, the strong annihilator graph is introduced with the vertex set , and two distinct vertices and are adjacent if and only if . It follows that each edge of is an edge of . In this paper, some properties such as connectivity, diameter, girth, etc of are studied and it is shown that if be a reduced ring, then is connected if and only if , and . Also, if be a non-reduced ring, then is connected with . Moreover, if R be a ring that is not an integral domain, then . Also, the perfectness and chordality of are studied. It is shown that if has finite clique number, then is chordal if and only if and is perfect if and only if , where is the set of all maximal ideals of .

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