On the zero-divisor graphs of finite free semilattices

SLx be the free semilattice on a finite nonempty set X. There exists an undirected graph Gamma(SLx) associated with SLx whose vertices are the proper subsets of X, except the empty set, and two distinct vertices A and B of Gamma(SLx) are adjacent if and only if A boolean OR B = X. In this paper, the diameter, radius, girth, degree of any vertex, domination number, independence number, clique number, chromatic number, and chromatic index of Gamma(SLx) have been established. Moreover, we have determined when Gamma(SLx) is a perfect graph and when the core of Gamma(SLx) is a Hamiltonian graph.