In recently years, many digital logic systems based on graph theory are analyzed and synthesized. This paper presented a method of constructing the function using edge valued extension graph which is based on graph theory. The graph is applied to a new data structure. from binary graph which is recently used in constructing the digital logic systems based on the graph theory. We discuss the mathematical background of literal and reed-muller expansion, and we discuss the edge valued extension graph which is the key of this paper. Also, we propose the algorithms which is the function derivation based on the proposed edge valued extension graph. That is the function minimization method of the n-variables m-valued functions and showed that the algorithm had the regularity with module by which the same blocks were made concerning about the schematic property of the proposed algorithm.