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Let R be a commutative ring with identity and be the set of nilpotent elements of R. The nil-graph of ideals of R is defined as the graph whose vertex set is , and there exists a nontrivial ideal J such that and two distinct vertices I and J are adjacent if and only if . Here, some graph properties of are studied. For instance, some bounds for the diameter, girth, and radius of are given. In case that is a finite graph, it is proved that the center and median of coincide. Furthermore, we determine when the edge chromatic number of equals its maximum degree. Also, for every ring R, it is shown that both the clique number and vertex chromatic number of equal , where n is the number of minimal prime ideals of R and t is the number of nonzero ideals of R which are contained in .