MGF 1107 – EXPLORATIONS IN MATHEMATICS LECTURE 13 Having now created some graphs from physical settings, we are now in a position to introduce some additional vocabulary. Definition: We say that two vertices are adjacent if they are joined by an edge. Ex. Definition: The degree of a vertex is the number of edges connected to it. Ex. Definitions: A vertex with an even number of edges connected to it is called an even vertex and a vertex with an odd number of edges connected to it is called an odd vertex. Ex. Definitions: A path is a sequence of adjacent vertices and the edges connecting them. The number of edges involved in a path is called the length. If a path starts and ends at the same point, we call it a circuit. Note: Every circuit is a path, but not every path is a circuit, since a circuit needs to start and end at the same vertex. It is important to recognize the difference between a graph and a path. A graph is a set of vertices and edges connecting them. A path can be thought of as movement from vertex to vertex. To represent a path we use a sequence of vertices separated by commas. Ex. Ex. (5.A.11) Ex. (5.A.13) Definitions: A graph is connected if for any two vertices in the graph, there is a path that connects them. So far all the graphs we have studied (except for the bowling league example) have been connected. If a graph is not connected we say it is disconnected. Ex. Definition: A bridge is an edge that if removed from a connected graph would make it disconnected. Ex. (5.A.15a) Note In the Königsberg bridge problem, none of the edges that represent real bridges are bridges in the graph-theory sense! Euler Paths and Euler Circuits Although we looked at the Königsberg bridge problem in the previous section, we didn’t actually solve it! Before we can do this we need to introduce two more definitions. Definition: An Euler path is a path that passes through each edge of a graph exactly one time. Definition: An Euler circuit is a circuit that passes through each edge of a graph exactly one time. The difference between the two is that an Euler circuit must start and end at the same vertex. Ex. For the graphs shown below, determine whether an Euler path or Euler circuit exists.