This chapter introduces the most commonly used graph notations that are referenced throughout the book. The chapter also covers the most important properties of graphs: diameter; clustering coefficient; degree distribution and average shortest path; and graph types and classifications, including random graphs, regular graphs, small-world graphs, and scale-free graphs. Finally, the chapter describes the representations that can be used in practical implementations, including adjacency lists and matrices.

Graph Terminology and Notations

A graph is a data structure consisting of a set of vertices connected by a set of edges that can be used to model relationships among the objects in a collection. Graphs are typically studied in the field of graph theory, which is an area of study in mathematical research.

Figure 1.1 shows the seven bridges in Königsberg that led to the invention of graph theory. The problem is to cross all bridges without crossing any bridge twice. Euler showed in 1741 that it was impossible to do so (Euler 1741).

Formally, a graph is defined as a set G = (V, E), where V is a collection of vertices V = {Vi, i = 1, n} and E is a collection of edges over V, Eij = {(Vi, Vj), Vi ∈ V, Vj ∈ V}. In alternative terminology, graphs also are referred to as networks, vertices as nodes, and edges as links.