This thesis gives presentation of two elementary graph algorithms called the breadth-first search (BFS) and the depth-first search (DFS). Many graph algorithms are organized as simple elaboration of the basic algorithms for searching graphs. Searching a graph means systematically following the edges of the graph so as to visit the vertices of the graph. The strategy followed by breadth-first search is to search “broader” in the graph. It computes the distance from s to each reachable vertex. It also produces a “breadth-first tree” with root s that contains all reachable vertices. The strategy followed by depth-first search is, as its name implies, to search “deeper” in the graph whenever possible and for each vertex v adjacent to vertex u recursively calls and checks the neighbors vertex v before returning and checking the remaining neighbor of vertex u. With algorithm descriptions, there are also their pseudocodes. It is proven that BFS finds shortest-path distance for a given vertex to each vertex in its component connections and there is also shown the use of DFS algorithm for topological sorting of directed acyclic graphs. For the purpose of this study, the application called "BFS" has been made. It calculates the shortest-path distance from a given vertex to all other vertex.