Graph theory, which is used effectively in many fields from science to liberal arts, has very important place in our lives. As a result of this, the topological structure of the graphs is studied by many researchers. In this paper, we investigate the topological spaces generated by the graphs. The states of being an accumulation point and an interior point of a point in these spaces are examined. It is defined that relative topology on a subgraph of a graph. It is shown that this topology is different from the topology generated by this subgraph. Moreover, using the minimal adjacencies of vertices set of a graph, necessary and sufficient conditions for being T_0-space, T_1-space and Hausdorff space of the topological space generated from this graph are presented. This enables to examine whether the topological space is T_0, T_1 and Hausdorff without obtaining the topology generated from the graph.