Abstarct: In this thesis,we present nonadabtive group testing. Also, three types of binary matrices have been found to be major tools in understanding and constructing pooling designs; we give their definitions. We shall show that nonadabtive group testing needs at least t group tests for identifying all the defective items from a group of t items having at most defective items. we use of graphs on group testing designs. Also we shall show that under restrictions on group size, optimal nonadabtive group testing can be constructed using Generalized Peterson Graphs. We present nonadabtive group testing algorithms, then those Generalize to error-tolerant model, presence of inhibitors, complex model and threshold model.We define new combinatorial structures with applications to efficient group testing with hinhibitors. Let N(d) denote the largest n for fixed d for which individual testing is optimal. In the rest of this thesis, we answer to query as follow; when is individual testing optimal for nonadabtive group testing? We shall show that N(d) = (d +1 )×2 for d =1 ,2 ,3 ,4 .