In this paper, we determine when (Γ I (L)) c , the complement of the zero divisor graph Γ I (L) with respect to a semiprime ideal I of a lattice L, is connected and also determine its diameter, radius, centre and girth. Further, a form of Beck's conjecture is proved for Γ I (L) when ω((Γ I (L)) c ) < ∞.

ABSTRACT. In this paper, it is proved that if B is a Boolean poset and S is a bounded pseudocomplemented poset such that S \ Z(S) = {1}, then Γ(B) ∼ = Γ(S) if and only if B ∼ = S. Further, we characterize the graphs which can be realized as zero divisor graphs of Boolean posets.

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In this paper, we study basic properties such as connectivity, diameter and girth of the zero-divisor graph [Formula: see text] of [Formula: see text] matrices over a lattice [Formula: see text] with 0. Further, we consider the zero-divisor graph [Formula: see text] of [Formula: see text] matrices over an [Formula: see text]-element chain [Formula: see text]. We determine the number of vertices, degree of each vertex, domination number and edge chromatic number of [Formula: see text]. Also, we show that Beck’s Conjecture is true for [Formula: see text]. Further, we prove that [Formula: see text] is hyper-triangulated graph.