On a Generalized Zero-Divisor Graph of a Lattice with respect to an Ideal

Let L be a lattice with the least element 0, and I be an ideal of L. In this paper, we introduce a generalized zero-divisor graph G (L) I Γ of L with respect to the ideal I. We show that G (L) I Γ is connected with diameter at most three. If G (L) I Γ has a cycle, we show that the girth of G (L) I Γ is at most four. We also investigate the existence of cut vertices of G (L) I Γ Moreover, we examine certain situations when G (L) I Γ is a complete bipartite graph.