Undirected Graphs
Graphs are composed of two components: vertices and edges.
Vertices are essentially points. (They are also referred to as
nodes.) Typically, they will be labeled on a graph.
In an undirected graph, edges are simply lines in between pairs
of vertices.
So, for example, in a graph with n vertices, the maximum
number of edges is
n
C
2
= n(n1)/2. This is the number of edges
in a complete graph. A complete graph is a graph where there
exists an edge between all pairs of vertices.
We will define the degree of each vertex of a graph to be the
number of edges that are incident to that vertex.
A
walk
in a graph is a sequence of edges that can be traversed
one by one. (This essentially means that the endpoint of an
edge in a path has to be the starting point of the next edge in
the path.) It is permissible for a walk to start and end in the
same place. (Or, of course, start and end in different places.)
A graph is connected if there exists a path in between all pairs
of vertices.
Here is an example of a undirected graph:
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Directed Graphs
The only difference between a directed graph and an
undirected one is how the edges are defined. In an undirected
graph, an edge is simply defined by the two vertices it connects.
But, in a directed graph, the “direction” of the edge matters.
For example, let’s say a graph has two vertices v and w. In a
directed graph with these two vertices, we would be allowed to
have more than one edge. We could have an edge from v TO w,
and another one from w TO v. In essence, each edge not only
connects a pair of vertices, but also has a direction associated
with it.
Using a formal definition, the set of edges of a directed graph
G can be defined as a subset of the Cartesian product V x V,
where V is the set of vertices.
All of the other definitions listed for undirected graphs apply
to directed ones.
Also, when we talk about the degree of a vertex in a directed
graph, we have to distinguish between the “in” degree and the
“out” degree. At each vertex, edges are either coming in or
going out. The in degree of a vertex is simply the sum of the
number of edge coming in to that vertex. The out degree is
defined similarly.
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 Spring '09
 Computer Science, Graph Theory, vertices

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