Graph Theory Lecture Notes 1
Example: Cocktail Party Graph
At any party of 6 people there must be a set of at least three people who are mutual friends
or a set of three people who are mutual strangers.
We can diagram this situation by representing the people as dots and joining two dots with a
red line if they are friends and a blue line if they are strangers. The statement can then be
rephrased as; in this diagram (no matter how the red and blue lines are drawn) there must be
either a red triangle or a blue triangle. Here is an example of such a diagram:
At any dot there are 5 lines. At least three must have the same color, for definiteness, let's
say that at a certain dot there are three red lines. If any two of the other endpoints of these
lines are joined by a red line, then there will exist a red triangle. If none of them are joined
by a red line, then all three are joined by a blue line, and so, there exists a blue triangle.
Example: Finite State Autonoma
We wish to build a simple machine (or circuit) which recognizes a specific pattern in a bit string and requires no memory (storage). The bit string is read one bit at a time. The following finite state autonoma recognizes bit strings that end in a 1. It consists of three states (start, accept and reject). Start in the start state and for each bit that is read, follow the arrow with that bit as a label to the next state. When the string is completely read, the state you are in tells you whether the string ended in a 1 (accept) or a 0 (reject).
Def: Graph, Digraph
Unless otherwise specified, our graphs (and digraphs) will have finite vertex sets, self-loops and
multiple edges (or arcs) are allowed. When no self-loops and multiple edges are present, we refer to the graph as
a simple graph.
Def: Adjacent Vertices (Edges) , Degree (valence) of a vertex, Degree sequence of a graph
Theorem 1.1.2 (Euler): The sum of the degrees of the vertices of a graph is twice the number of edges.
- The number of vertices of odd degree is even.
- The sum of the degree sequence of a graph is even.
Def: Regular Graphs
Here are some some famous little simple graphs that I have known:
- K4 : The complete graph on 4 vertices. The K stands for Kuratowski, a pioneer in the area of
graph theory. The term complete refers to the fact that all the possible edges are present. In
general, the complete graph on n vertices is denoted Kn, and is an (n-1)-regular simple graph.
- K3,3 : A complete bipartite graph. (The K still stands for Kuratowski.) This is also known as
the 3-utilities graph, or the water-light-gas graph. Bipartite refers to the
fact that the vertices can be grouped into two sets, with no edges existing between vertices in
the same set. The term complete here means that all possible edges between the two sets
exist. (In general, the two sets need not have the same size). The common name for this
graph comes from a famous old problem (involving houses and wells) which has been
updated. The problem now is stated as: there are three houses and three utility company
supply terminals (the water company, the gas company and the electric company). Each utility
needs to run a supply line from the terminal to each of the houses. Can these supply lines be
situated so that no two of them cross each other (except, of course, at the houses). This
problem (whose answer is no), leads to the interesting concept of planarity of graphs.
- C4 : The cyclic graph on 4 vertices. In general Cn refers to the cyclic graph on n vertices and is a 2-regular simple graph. Cn is bipartite when n is even.
- Petersen's Graph : This graph on 10 vertices and 15 edges is very famous because it tends
to be a counter-example to many generalizations of ideas that work for smaller graphs. As a
rule of thumb, check any conjecture on the Petersen graph before trying to prove it.
- Hypercube Graphs : A hypercube graph Qn is the n-regular graph whose vertex set is the set of bitstrings of length n, and such that there is an edge between two vertices iff they differ in exactly one bit. The diagram above pictures Q4.
- Circular Ladder Graphs: The circular ladder graph CLn consists of two concentric n-cycles in which each of the n corresponding vertices is joined by an edge. It is a 3-regular simple graph. The diagram above pictures CL8.