A graph is composed of vertices and edges. Vertices are connected by edges. A graph can be directed or undirected. In a directed graph, edges have direction but in an undirected graph edges do not have direction. The degree of a vertex is the number of edges connected to it. A path in a graph is a sequence of adjacent vertices and a cycle is a path from a vertex back to itself. Connectedness refers to whether there is a path between all pairs of vertices in an undirected graph. Graphs can be represented using adjacency matrices or adjacency lists.

1. Graph – Basic concepts

2. Basic Concepts
A graph is an ordered pair (V, E).
V is the set of vertices. (You can think of them as integers 1,
2, …, n.)
E is the set of edges. An edge is a pair of vertices: (u, v).
Edges can be labeled with a weight:
6-Graphs
10

3. Concepts: Directedness
In a directed graph, the edges are “one-way.” So an edge (u, v)
means you can go from u to v, but not vice versa.
In an undirected graph, there is no direction on the edges: you can go
either way. (Also, no self-loops.)
6-Graphs
a self-loop

4. Concepts: Adjacency
Two vertices are adjacent if there is an edge between them.
For a directed graph, u is adjacent to v iff there is an edge (v, u).
6-Graphs
u w
v
u is adjacent to v.
v is adjacent to u and w.
w is adjacent to v.
u w
v
u is adjacent to v.
v is adjacent to w.

5. Concepts: Degree
Undirected graph: The degree of a vertex is the number of edges
touching it.
For a directed graph, the in-degree is the number of edges entering
the vertex, and the out-degree is the number leaving it. The degree is
the in-degree + the out-degree.
6-Graphs
degree 4
in-degree 2, out-degree 1

6. Concepts: Path
A path is a sequence of adjacent vertices. The length of a path is the
number of edges it contains, i.e. one less than the number of vertices.
We write u ⇒ v if there is path from u to v. We say v is reachable
from u.
6-Graphs
1
2
3
4
Is there a path from 1 to 4?
What is its length?
What about from 4 to 1?
How many paths are there from 2 to
3? From 2 to 2? From 1 to 1?

7. Concepts: Cycle
•A cycle is a path of length at least 1 from a vertex to itself.
•A graph with no cycles is acyclic.
•A path with no cycles is a simple path.
•The path <2, 3, 4, 2> is a cycle.
6-Graphs
1
2
3
4

8. Concepts: Connectedness
•An undirected graph is connected iff there is a path between any two
vertices.
•The adjacency graph of U.S. states has three connected components.
Name them.
•(We say a directed graph is strongly connected iff there is a path
between any two vertices.)
6-Graphs
An unconnected graph with three
connected components.

9. ADJACENCY MATRIX
REPRESENTATION
• 1 2 3 4
1 1 1 1 1
2 1 0 0 0
3 0 1 0 1
4 0 1 1 0

10. ADJACENCY LIST
REPRESENTATION
 1 -> 1 -> 2 -> 3 -> 4
2 -> 1
3 -> 2 -> 4
4 -> 2 -> 3

11. GUIDED READING

13. ASSESSMENT
1. Which of the following statements is true?
A. A graph can drawn on paper in only one
way.
B. Graph vertices may be linked in any
manner.
C. A graph must have at least one vertex.
D. A graph must have at least one edge.


14. CONTD..
2. Suppose you have a game with 5 coins in
a row and each coin can be heads or tails.
What number of vertices might you expect to
find in the state graph?
A. 7
B. 10
C. 25
D. 32

15. CONTD..
3. Why is the state graph for tic-tac-toe a
directed graph rather than an undirected
graph?
A. Once a move is made, it cannot be
unmade.
B. There is an odd number of vertices.
C. There is an odd number of edges.
D. There is more than one player in the
game.

16. CONTD..
4. A simple graph has no loops. What other
property must a simple graph have?
A. It must be directed.
B. It must be undirected.
C. It must have at least one vertex.
D. It must have no multiple edges.

17. CONTD..
5. Suppose you have a directed graph
representing all the flights that an airline flies.
What algorithm might be used to find the best
sequence of connections from one city to
another?
A. Breadth first search.
B. Depth first search.
C. A cycle-finding algorithm.
D. A shortest-path algorithm.