3-Zero-Divisor Hypergraph with Respect to an Element in Multiplicative Lattice

Abstract

Let ?? be a multiplicative lattice and ?? be a proper element of ??. We introduce the 3-zero-divisor hypergraph of ?? with respect to ?? which is a hypergraph whose vertices are elements of the set {??1????{??}|??1??2??3??????1??2???,??2??3??? ?????? ??1??3??? ?????? ???????? ??2,??3????{??}} where distinct vertices ??1,??2 and ??3 are adjacent, that is, {??1,??2,??3} is a hyperedge if and only if ??1??2??3??????1??2???,??2??3??? ?????? ??1??3???. Throughout this paper, the hypergraph is denoted by ??3(??,??). We investigate many properties of the hypergraph over a multiplicative lattice. Moreover, we find a lower bound of diameter of ??3(??,??) and obtain that ??3(??,??) is connected.