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.

Cops and robbers game is a game usually played on a finite connected graphwith two players, cop and robber. Recently, cops and robbers game played on hypergraphs was introduced. To give a better chance to a cop by allowing morethan one cop and at least one cop has to move, the cop-number, the least numberof cops to guarantee that they win the game, on graphs and hypergraphs is studied.This thesis provides (i) a characterization of a cop-win hypergraph (ii) some results on the products of hypergraphs and (iii) the cop-number of complete k-partite hypergraphs and n-prisms over a hypergraph. Moreover, the cop-number of a special class of graphs is determined.