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In this paper, we determine when 
$\mathop{({\Gamma }_{I} (L))}\nolimits ^{c} $, the complement of the zero divisor graph 
${\Gamma }_{I} (L)$ 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 
${\Gamma }_{I} (L)$ when 
$\omega (\mathop{({\Gamma }_{I} (L))}\nolimits ^{c} )\lt \infty $.
