Abstarct: In this dissertation, we analyze the coloring, the number of cliques and the edge independence number and edge covering number in the total graphs on the Mycielski graphs and central graphs. For this purpose, we first calculate the achromatic number of the central graph, middle graph and total graph on the star graphs and the equitable chromatic number of the central graphs on the star graphs, complete bipartite graphs and complete graphs and also total graph of paths and cycles. Then, we find the number of triangles for the total graph, middle graph, central graph, Mycielski graph and some of their combinations and we provide upper bounds for the number of triangles of the n−iterated Mycielski graph of a graph and the total graph of the n−iterated Mycielski graph of it. Moreover we obtain the edge independence number and edge covering number of the total graph of the Mycielski graphs on the star graphs and some of the trees in terms of the vertex and edge independece number and the vertex and edge covering number. In addition we calculate the vertex and edge independence number and vertex and edge covering number of the central graphs, middle graphs and total graphs on the caterpillar. Finally, we present our new results on the achromatic coloring and find the achromatic number of the central graph on the paths, cycles, bipartite graphs and caterpillar graphs.