On topological structures of the simple undirected graphs

Graph theory is a branch of mathematics. The theory has also a very important place in our

lives, since the graphs are used effectively in many fields. As a result of this, the topological

structure of the graphs has been a interesting research subject. Many researchers has studied on

this subject. In this paper, we investigate some topological notions such as accumulation point,

interior point, relative topology, T0,T1 and Hausdorff space on the topological spaces generated

the graphs. Firstly, it is examined that the states of being an accumulation point and an interior

point of a point in these spaces. Relative topology on a subgraph of a graph is defined. It is shown

that this topology is not equal to the topology generated by this subgraph. Finally, necessary and

sufficient conditions for being T0−space, T1−space and Hausdorff space of the topological space

generated from this graph are presented. This enables to examine whether the topological space is

T0,T1 and Hausdorff without obtaining the topology generated from the graph .