1.  Consider the cities Atlanta, Boston, Chicago, Detroit, Fargo, and Los Angeles.  We want to see what connections there are between cities on a particular Tuesday.

2.  United: Atlanta-Boston, Atlanta- Chicago, Boston-LA, Chicago-LA  American: Atlanta-Detroit, Atlanta- LA, Chicago-LA, Detroit-LA  Southwest: Atlanta-Chicago, Boston- Chicago, Chicago-Detroit, Chicago-LA  Ignoring the airline and direction of the flights, draw a picture that represents the situation.

3.   A graph is a set of points (called vertices, or nodes) and a set of lines called edges connecting some pairs of vertices.   Two vertices connected by an edge are said to be adjacent.   Vertices may be connected by more than one edge; A vertex need not be connected to any other vertex; A vertex may be connected to itself.   The degree of a vertex is the number of edges adjacent to it, i.e. the number of connections in which it is involved.

4. In Groups  For any graph, what can you say about the sum of the degrees of all the vertices?  Discuss.

5. You can’t get there from here   Even if there isn’t a direct route, we would like to know if it is possible to get from one place to the next.   Can we get from Boston to Detroit today?   Can we get from Chicago to Fargo today?   While these questions are relatively easy to answer for a small graph, as the number of vertices and edges grows, it becomes harder to keep track of all the different ways the vertices are connected.

6. The MATRIX  A matrix is a rectangular array of items—in our case, numbers.  Matrix operations

7.   Matrix notation and computation can help to answer the graph questions.   The adjacency matrix for a graph with n vertices is an n x n matrix whose (i, j) entry is 1 if the i th vertex and j th vertex are connected, and 0 if they are not.   Write the adjacency matrix for the Tuesday flight example.