On Graph Theory Based on Matrix

Abstract

This paper presents a new approach of graph theory based on natural numbers and matrix only. The natural number is used for vertex and matrix for the edge. The fast graph generation is an advantage of this approach by the successor function. The connected and non-connected graph both is defined by the axiom. There is the arithmetic based on the additive and multiplicative operation over the matrix with the set of natural number. Basically, the graph is defined here without any structure or geometry. This new approach lies with the set of natural numbers and its matrix representation. The proposed graph theory is based on axioms. The geometrical phenomenon is the key inspiration of this theory. In conventional graph theory, the distance between two vertices is defined only by the straight line. The curve distance is ignored. In the proposed graph theory, the straight line as well as curve are also considered. The matrix representation is sufficient for both the edges. If there will be knot instead the curve, then the defined representation is formed the graph. Thus, it is associated with smaller domain and range than the existed graph theory.