Abstract
Let R be a commutative ring with nonzero identity and k≥2 be a fixed integer. The k-zero-divisor hypergraph Hk(R) of R consists of the vertex set Z(R,k), the set of all k-zero-divisors of R, and the hyperedges of the form {a1,a2,a3,…,ak}, where a1,a2,a3,…,ak are k distinct elements in Z(R,k), which means (i) a1a2a3⋯ak=0 and (ii) the products of all elements of any (k−1) subsets of {a1,a2,a3,…,ak} are nonzero. This paper provides two commutative rings so that one of them induces a family of complete k-zero-divisor hypergraphs, while another induces a family of k-partite σ-zero-divisor hypergraphs, which illustrates unbalanced or asymmetric structure. Moreover, the diameter and the minimum length of all cycles or girth of the family of k-partite σ-zero-divisor hypergraphs are determined. In addition to a k-zero-divisor hypergraph, we provide the definition of an ideal-based k-zero-divisor hypergraph and some basic results on these hypergraphs concerning a complete k-partite k-uniform hypergraph, a complete k-uniform hypergraph, and a clique.
Highlights
Graph structures and algebraic structures are closely related
Ideal-Based k-Zero-Divisor Hypergraphs In the previous consideration, we considered a commutative ring D I, where D is a principal ideal domain (PID) and I is the appropriate ideal of D, which enabled us to define the relationship between an algebraic structure and a hypergraph structure related to an ideal-based zerodivisor graph
In addition to a complete k-partite k-uniform hypergraph, we investigate the completeness of HkI (R)
Summary
Graph structures and algebraic structures are closely related. For example, if R is a commutative ring and Z(R) is the finite set of all zero-divisors of R, Z(R) can be regarded as a set of vertices of a graph G, and two elements in Z(R) can have an edge connecting between them whenever their product is zero. The vertex set consists of all nonzero zero-divisors of commutative rings They investigated the completeness and automorphisms, denoted by Aut(Γ R) , of a zero-divisor graph. A k-zero-divisor hypergraph of a commutative ring R with nonzero identity, denoted by Hk(R), is defined as a k-uniform hypergraph H whose vertex set is Z(R, k), and the set {a1, a2, a3, . They found that there exists a complete k-partite k-zero-divisor hypergraph whose vertex set is Z D Dγ, k , where γ = ∏ik=1 pi, k ≥ 2, and D is a principal ideal domain (PID) with nonzero identity containing at least distinct k prime elements, say p1, p2, p3, .
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