Abstract
This chapter explores the object-oriented graph theory. A graph consists of a finite set of vertices, some of which are joined by edges. Mathematically, a graph can be considered as a relation between vertices. Two vertices are related if and only if they are joined by an edge. There are three different ways to represent graphs, which require operations translating between them; the whole situation can be embedded in a small hierarchy of classes consisting of an abstract top class graph, under the class Object, followed by three subclasses, one for each way of representing graphs. These three subclasses are where the graphs are actually created. The default method for processing in the class graph is NIM [self, <name of method>]. The meaning of this is that these methods must be implemented in the subclasses of graph. If they are not, then a message to that effect is returned. Products are operations on graphs that depend on more than one graph. In Mathematica, functional programs for graph construction can be written depending on several graphs, provided they are accessed through other methods to which they can respond. Two notable products of graphs are Cartesian products and Tensor products. There are thousands of algorithms that use graphs in one way or another; in principle, such algorithms belong in the class graph.
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