Graphs are among the most ubiquitous models of both natural and human-made structures. They can be used to model many types of relations and process dynamics in computer science, physical, biological and social systems. Many problems of practical interest can be represented by graphs. In general graphs theory has a wide range of applications in diverse fields. This paper explores different elements involved in graph theory including graph representations using computer systems and graph-theoretic data structures such as list structure and matrix structure. The emphasis of this paper is on graph applications in computer science. To demonstrate the importance of graph theory in computer science, this article addresses most common applications for graph theory in computer science. These applications are presented especially to project the idea of graph theory and to demonstrate its importance in computer science.