1. Graph Theory

2. Objectives
• Understand the terms:
– Graph
• Vertex
• Arc
– Simple Graph
– Directed Graph
– Adjacency Matrix
– Adjacency List

3. Definitions
• Graph is a finite number of points
connected by lines. Points are
normally called vertices or nodes
• Lines are called edges or arcs
Edge/Arc
Vertex
Node

4. Network or weighted graph
• Each edge/arc has an associated
number
5
Time
Distance
4
Money
3
3

5. “Connected”
• Vertices are connected if there is a edge
joining them
• A graph is connected if all pairs of vertices
are connected
• A Simple Graph is one in which there are no
loops and at most one edge connects any
pair of vertices
• A degree (or order) of a vertex is the number
of edges connected to the vertex

6. Worked examples
• Simple Graph G has six vertices and
their degrees are
2d,2d,2d+1,2d+1,2d+1,3d-1 – where d
is an integer
• Show that d is even
• Use the fact that the graph is simple to
show that d < 3 and find a value for d
• Draw a possible graph G

7. Directed graphs
• Graph that has directed edges, eg:
arrows on the edges
8 12
10
15

8. Complete Graph
• Every vertex is connected by an edge
to each of the other vertices.
5 vertices
4+
3+
2+
1 edges

9. Question
• Graph G has four vertices and edges
of length 7,8, 8 and 9
• Explain why G is not a complete graph
• Stat the number of edges that must be
added to G to make it complete
• Draw a directed graph for a round-
robin tournament involving three
teams - A, B and C

10. Adjacency Matrix
x 1 2 3 4 5
1 3 1 - 1 1 1 0
5 2 1 - 0 1 0
3 1 0 - 1 2
2 4
4 1 1 1 - 1
5 0 0 2 1 -

11. Adjacency List
Vertex Adjacent
Vertices
1 3
1 2,3,4
5
2 1,4
2 4
3 1,4,5
4 1,2,3,5
5 3,4

12. Objectives
• Understand the terms:
– Graph
• Vertex
• Arc
– Simple Graph
– Directed Graph
– Adjacency Matrix
– Adjacency List