MT4514: Graph Theory

Chapter 2Basic definitions and isomorphismIn this chapter we give some of the formal definitions of graph theory, and establishsome elementary properties of graphs.1. Some definitionsDefinition 1.1. A simple graph G consists of a finite set V of vertices (also calledpoints or nodes) and a set E of edges such that each edge joins a pair of vertices inV . More formally, a simple graph is a pair (V, E) where V is a finite set and E isa set of subsets of V , each of size 2.When V and E are finite sets we write |G| or |V | to denote the number ofvertices of G. The graph G is finite when V and E are finite. We denote the edgejoining v i and v j by v i v j . Note that v i v j = v j v i as the edges of a graph do not havea direction.Example 1.2. Here are some finite and infinite graphs:Definition 1.3. A loop is an edge joining a vertex to itself. A graph has multipleedges if there exist two vertices with more than one edge between them. In general,a graph may have loops and multiple edges: a simple graph is a finite graph withno loops and no multiple edges.Often we will simply say “graph” for “finite graph” when the context is clear.Definition 1.4. A graph is connected if it is possible to go between any pair ofvertices by traversing a sequence of edges. If a graph G is not connected then wemay consider G as a disjoint union of connected graphs, known as the connectedcomponents of G.Example 1.5. The graph G is connected, the graph H is unconnected and has twoconnected components.8