Graph theory means the study of graphs . It is one of the important branch of mathematics. It is considered as one of the powerful tool to solve the large complex problems. Graph theory is considered as one of the prime object of study discrete mathematics. The theory of graph is originated from the famous Konigsberg seven bridges problem which was solved by great mathematician Euler. The term “graph” was introduced by Sylvester in a paper published in 1878 A graph in this context is formed by set of vertices(also called nodes or points) and edges(also called arc or lines). An edge is formed by two vertices. They are used to show relationship between two nodes or objects. Many real life problems can be represented using graph. In computer science graphs are used to represent network, to organize data and to show flow of system. For eg. The traversing path of any website can be represented by best use of directed graph. Graph theory supports many algorithms to smoothen the task of problem solving. One of the famous problem in graph theory is graph coloring problem. Graph theory also provides solutions to problem of chemistry, economics, probability, biology, geometry etc. In this course we introduce some of the basic terminology of graph theory and then we study some important graphs and operations on the graphs.