The study of graph theory and its applications have increased substantially in the past 40 years. This is especially the case in the applications to the fields of computer science, electrical, and industrial engineering. Applications vary from designing computer software and hardware to telephone networks to airline network.In the thesis, properties are studied in detail to include definitions, theorems, examples and implementations of four graph algorithms in two important topics, namely connectivity and shortest paths. In the implementation part, algorithms will be available to solve problems on selected graphs. A graphic representation is also included to have a better picture of a problem and its solution.