In this first part of the book we develop some of the basic ideas behind graph theory – the study of network structure. This approach allows us to formulate basic network properties in a unifying language. The central definitions discussed here are simple enough that we can describe them relatively quickly at the outset; after this, we consider some fundamental applications of the definitions.

Basic Definitions

Graphs: Nodes and Edges. A graph is a way of specifying relationships among a collection of items. A graph consists of a set of objects, called nodes, with certain pairs of these objects connected by links called edges. For example, the graph in Figure 2.1(a) consists of four nodes labeled A, B, C, and D; node B is connected to each of the other three nodes by edges, and nodes C and D are also connected by an edge. We say that two nodes are neighbors if they are connected by an edge. Figure 2.1 shows the typical way to draw a graph: a small circle represents each node, and a line connects each pair of nodes that are linked by an edge.

When looking at Figure 2.1(a), think of the relationship between the two ends of an edge as being symmetric; the edge simply connects them to each other. In many settings, however, we want to express asymmetric relationships – for example, that A points to B but not vice versa.