We consider zero-divisor graphs with respect to primal, nonprimal, weakly prime and weakly primal ideals of a commutative ring R with non-zero identity. We investigate the interplay between the ringtheoretic properties of R and the graph-theoretic properties of ${Gamma}_I(R)$ for some ideal I of R. Also we show that the zero-divisor graph with respect to primal ideals commutes by localization.