MT4514: Graph Theory

Graph Theory 43(3) ⇒ (4): The removal of an edge from G produces a graph H with v −2 edges.We prove by induction that any graph with v − 2 edges is disconnected. If v = 2then the result is clear, so assume inductively that the result holds for all graphs onless than v vertices. The sum of the vertex degrees in H is 2(v − 2). If H has anyvertices of degree 0 then H is disconnected and we are done, so all vertices in H areincident to at least one edge. Since 2v > 2(v − 2) there exists at least one vertex, wsay, of H that has degree 1. Consider the graph H ′ produced by removing w andthe edge ending at w. This is a graph with v − 1 vertices and v − 3 edges, so by theinductive hypothesis H ′ is disconnected. Therefore H is disconnected.(4) ⇒ (5): Since G is connected there exists at least one path from any vertex toany other. If two vertices were connected by more than one path, then it would bepossible to remove an edge from G, lying on one of the paths, without disconnectingG. Therefore any two vertices are connected by exactly one path.(5) ⇒ (6): If G contained a circuit, then any pair of vertices on the circuit wouldbe connected by at least two chains. If an edge ab is added to G then a circuit willbe created, since there is already a path from a to b. If more than one circuit iscreated then there must be more than one path from a to b in G, a contradiction.(6) ⇒ (1): If G is disconnected, then the addition of an edge between onecomponent of G and another would not create a circuit, a contradiction. ThereforeG is connected and contains no circuits, so is a tree. A vertex of degree 1 is called an endpoint.Corollary 1.3.1. Every nontrivial tree has at least two endpoints.2. Let G be a forest with n vertices and k components, then G has n − k edges.Exercise 1.4. Prove this corollary.2. Counting TreesWe have already seen that people are interested in counting graphs. Whilst in generalgraph counting problems can be very hard, if we restrict ourselves to countingtrees then there are strong results. In this section we present a famous result byCayley in 1889 on the number of labelled trees.Definition 2.1. A labelled graph is a pair (G, φ) where G is a graph on v verticesand φ is a bijection from the vertices of G to the set {1, . . . , v}. Less formally,we associate to each vertex of G a distinct number from {1, . . . , v}. Two labelledgraphs (G 1 , φ 1 ) and (G 2 , φ 2 ) are isomorphic if there is a graph isomorphism fromG 1 to G 2 that preserves the labelling of the vertices.Example 2.2. The first of these graphs is labelled, the second is not.The first two of these labelled graphs are isomorphic as labelled graphs. The thirdgraph is isomorphic as a graph to the first two, but not as a labelled graph sincethe vertex of degree 4 has a different label.