Graphs, Paths and Circuits

• The good people of Konigsberg wondered if they could walk through their town, crossing all bridges exactaly once.
  • 
  • Euler produced graphs as a tool to study this.
• A graph contains
  • A finite set of points called vertices (or a vertex)
  • A set of line segment between the vertices called edges.
• Graphs can be used to represent
  • Social Networks
  • roads (intersections are the vertices, open roads are the edges.
  • Families (each person is a vertex, relationship (mom, dad, ...) are edges.
  • Airline flights
• A loop is a edge from a vertex the same vertex.
• The degree of a vertex is the number of edges connected to that vertex.
  • A vertex with even degree is an even vertex
  • A vertex with odd degree is an odd vertex
• A path is a sequence of adjacent vertices and the edges connecting them.
• A simple path is a path where not verticies are repeated.
• A graph is connected if there for any two verticies there is a path connecting them.
• A graph which is not connected is disconnected.
• A circuit is a path that begins and ends at the same vertex.
• A bridge is an edge which, if removed, would create a disconnected graph.
• Do problems 9-15 page 926
• Do problems 16-20
• Do problems 21-40