Graphs have provided a powerful methodology to solve a lot of real-world problems, and therefore there have been many proposals on the graph representations and algorithms. But, because most of them considered only memory-based graphs, there are still difficulties to apply them to large-scale problems. To cope with the difficulties, this paper proposes a graph representation and graph algorithms based on the well-developed relational database theory. Graphs are represented in the form of relations which can be visualized as relational tables. Each vertex and edge of a graph is represented as a tuple in the tables. Graph algorithms are also defined in terms of relational algebraic operations such as projection, selection, and join. They can be implemented with the database language such as SQL. We also developed a library of basic graph operations for the management of graphs and the development of graph applications. This database approach provides an efficient methodology to deal with very large- scale graphs, and the graph library supports the development of graph applications. Furthermore, it has many advantages such as the concurrent graph sharing among users by virtue of the capability of database.