Graphs and Trees
This handout:
• Terminology of Graphs
• Applications of Graphs
The process of mathematical reasoning
Derive properties,
get applications
Using the tools
of logical reasoning
Mathematical
objects
numbers
sets
functions
graphs
• We considered the first three types of mathematical objects
• Next: Graphs, their properties and applications
Terminology of Graphs
• A graph (or network) consists of
– a set of points
– a set of lines connecting certain pairs of the points.
The points are called nodes (or vertices).
The lines are called arcs (or edges or links).
• Example:
Graphs in our daily lives
•
•
•
•
•
•
Transportation
Telephone
Computer
Electrical (power)
Pipelines
Molecular structures in biochemistry
Terminology of Graphs
• Each edge is associated with a set of two nodes, called its endpoints.
Ex: a and b are the two endpoints of edge e
• An edge is said to connect its endpoints.
Ex: Edge e connects nodes a and b.
• Two nodes that are connected by an edge are called adjacent.
Ex: Nodes a and b are adjacent.
a
e
f
b
c
Terminology of Graphs: Paths
• A path between two nodes is a sequence of distinct
nodes and edges connecting these nodes.
Example:
a
b
• Walks are paths that can repeat nodes and arcs.
A little history:
the Bridges of Koenigsberg
• “Graph Theory” began in 1736
• Leonhard Eüler
– Visited Koenigsberg
– People wondered whether it is possible to take a
walk, end up where you started from, and cross
each bridge in Koenigsberg exactly once
The Bridges of Koenigsberg
A
1
2
3
B
5
4
C
6
7
D
Is it possible to start in A,
cross over each bridge exactly once,
and end up back in A?
The Bridges of Koenigsberg
A
1
2
3
B
5
4
C
6
7
D
Translation into a graph problem: Land masses are “nodes”.
The Bridges of Koenigsberg
A
1
2
3
B
5
4
6
D
C
7
Translation into a graph problem : Bridges are “arcs.”
The Bridges of Koenigsberg
A
1
2
3
B
5
4
6
D
C
7
Is there a “walk” starting at A and ending at A and
passing through each arc exactly once?
Such a walk is called an eulerian cycle.
Adding two bridges creates such a walk
A
1
8
2
B
4
5 6
3
C
9
D
7
Here is the walk.
A, 1, B, 5, D, 6, B, 4, C, 8, A, 3, C, 7, D, 9, B, 2, A
Note: the number of arcs incident to B is twice the
number of times that B appears on the walk.
Existence of Eulerian Cycle
4
A
1
6
8
2
B
4
5 6
3
C
4
The degree of
a node is the
number of
incident arcs
9
D
7
4
Theorem. An undirected graph has an eulerian
cycle if and only if
(1) every node degree is even and
(2) the graph is connected (that is, there is a path
from each node to each other node).
Graph properties
• Definition: The total degree of a graph is the sum of the
degrees of all its nodes.
• Theorem: If G is any graph, then the total degree of G
equals twice the number of edges of G:
the total degree of G = 2 (the number of edges of G)
• Corollary 1: The total degree of a graph is even.
• Corollary 2: In any graph there are an even number of
vertices of odd degree.
• Application to an Acquaintance Graph:
Is it possible in a group of five people
for each to be friends with exactly three others?
Terminology of Graph: Paths
• A path between two nodes is a sequence of distinct
nodes and edges connecting these nodes.
Example:
a
b
• Two nodes are called connected if there is a path
between them.
• Fact: For any two nodes a and b of a graph, there is
an efficient way to determine whether a and b are
connected or not.
An application of graphs
in solving a puzzle
From an initial position on the left bank of a river,
a ferryman wants to transport
a wolf, a goat, and a cabbage to the right bank.
Ferryman’s boat is only big enough
to transport one object at a time, other than himself.
For obvious reasons,
• the wolf cannot be left alone with the goat;
• the goat cannot be left alone with the cabbage.
How should the ferryman proceed?
An application of graphs in solving a puzzle
To solve the puzzle, create the following graph:
 Create a node for each allowable arrangement.
E.g., ( fg | wc ) is an allowable arrangement
since the ferryman and the goat are on the left bank,
and the wolf and the cabbage are on the right bank.
 Create an edge between two nodes if it is possible to go
from the arrangement of one node to the arrangement of the
other node by a single ferry trip.
E.g., there is an arc between nodes ( fgw | c ) and ( w | fgc ) because
the transition from the first node to the second node can be realized
by a single trip of the ferryman with the goat
from the left bank to the right bank.
An application of graphs in solving a puzzle
The resulting graph is:
fwgc |
fwg | c
fwc | g
wc | fg
w | fgc
g | fwc
fgc | w
c | fwg
fg | wc
| fwgc
To transport everything from the left bank to the right bank, we need to
find a path from node ( fwgc | ) to node ( | fwgc ) in the graph.
There are two this kind of paths. One of them:
(fwgc | )  (wc | fg)  (fwc | g)  (w | fgc)  (fwg | c) 
(g | fwc)  (fg | wc)  ( | fwgc)