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(Oct. 15) k-Zero-Divisor Hypergraphs

Last updated :2018-10-08

Topic: k-Zero-Divisor Hypergraphs
Speaker: Dr. Pinkaew Siriwong
(Chulalongkorn University, Thailand)
Time: 15:00-17:00, Monday, October 15, 2018
Venue: Room 416, Mathematics Building, Guangzhou South Campus, SYSU

Abstract:
Graph structures and algebraic structures are related; that is, a zero-divisor graph. In master thesis, we generalized the idea of a zero-divisor graph into a k-zero-divisor hypergraph including the vertex set Z(R,k), the set of all k-zero-divisors of R where k>=2. A subset {a1, a2, a3,…, ak} of Z(R, k) is an (hyper)edge if and only if (i) a1a2a3 …ak = 0 and (ii) the products of all elements of any (k 􀀀 1)-subsets of {a1, a2, a3,…, ak} are nonzero. We provided (i) a necessary condition of commutative rings that implies the completeness of their k-zero-divisor hypergraphs; (ii) a necessary condition of commutative rings that implies the ability to partition their set of all k-zero-divisors into k partite sets and the completeness of that k-partite k-zero-divisor hypergraphs; and (iii) a necessary condition of commutative rings that implies the ability to partition their set of all σ-zero-divisors into k partite sets, for some integer σ>=k. Moreover, we determined its diameter and minimum length of all cycles. Recently, we have been interested in the vertex-pursuit game played on hypergraphs.