MT4514: Graph Theory

Graph Theory 9Definition 1.6. Two vertices v 1 , v 2 ∈ V are adjacent if there is an edge joining v 1to v 2 . The degree or valency of a vertex v is the number of edges that have one endat v: each loop at a vertex v contributes 2 to the valency of v. The graph is regularif all vertices have the same degree.Example 1.7. This graph has vertices labelled by their valencies.Lemma 1.8 (Handshaking lemma) The sum of all of the degrees of all verticesof a finite graph is even.Proof. This is clear since each edge contributes 1 to the degree of two vertices,or 2 to a single one. Definition 1.9. A graph G is planar if G can be drawn in the plane (i.e. on a flatpiece of paper) without the edges crossing.Example 1.10. G is a planar graph, H is not.2. IsomorphismThe way that we draw a graph, or the nature of the set V , does not matter when weare studying graphs. Consider the bijection φ between the following graphs, whichshows that they are essentially the same: