 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.
 United:
Atlanta-Boston, AtlantaChicago, Boston-LA, Chicago-LA
 American: Atlanta-Detroit, AtlantaLA, Chicago-LA, Detroit-LA
 Southwest: Atlanta-Chicago, BostonChicago, Chicago-Detroit, Chicago-LA
 Ignoring
the airline and direction of
the flights, draw a picture that
represents the situation.
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.

In Groups
 For
any graph, what can you say
about the sum of the degrees of all
the vertices?
 Discuss.
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.

The MATRIX
A
matrix is a rectangular array of
items—in our case, numbers.
 Matrix operations
 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 ith vertex
and jth vertex are connected, and 0 if
they are not.
 Write the adjacency matrix for the
Tuesday flight example.