The.Algorithm.Design.Manual.Springer-Verlag.1998

The Friendship Graph
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The Friendship Graph
Figure: A portion of the friendship graph
To demonstrate the importance of proper modeling, let us consider a graph where the vertices are people,
and there is an edge between two people if and only if they are friends. This graph is well-defined on any
set of people, be they the people in your neighborhood, at your school or place of business, or even the
entire world. There are many interesting aspects of people that are best understood as properties of this
friendship graph.
We use this opportunity to define important graph theory terminology. ``Talking the talk'' proves to be an
important part of ``walking the walk''.
● If I am your friend, does that mean you are my friend?
A graph is undirected if edge (x,y) always implies (y,x). Otherwise, the graph is said to be
directed. The ``heard-of'' graph is directed, since many famous people who I have heard of have
never heard of me! The ``had-sex-with'' graph is presumably undirected, since the critical
operation always requires a partner. I'd like to think that the ``friendship'' graph is always an
undirected graph.
● Am I my own friend?
An edge of the form (x,x) is said to be a loop. If x was y's friend several times over, we can model
this relationship using multiedges, multiple edges between the same pair of vertices. A graph is
said to be simple if it contains no loops and no multiple edges. Simple graphs really are often
simpler to work with in practice. Therefore, we might be better off if no one was their own friend.
● How close a friend are you?
A graph is said to be weighted if each edge has an associated numerical attribute. We could model
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