1. Introduction to Graph
 A graph consists of a set of vertices, and a set of edges that link together the vertices. 
A graph can be:
Directed: Edges are directed.
Undirected: Edges are undirected
Data Structure and Algorithm

2. Undirected Graph A graph G = (V, E) V: vertices
E : edges, unordered pairs of vertices from V  V
(u,v) is same as (v,u)
Thus |E| <= |V| (|V|-1)/2 A O N M L K J E F G H D C B I P
Data Structure and Algorithm

3. Directed Graph A graph G = (V, E) V: vertices
E : edges, ordered pairs of vertices from VV
(u,v) is different from (v,u)
Thus |E| <= |V| (|V|-1) A E F G H D C B I
Data Structure and Algorithm

4. Graph in Application Internet: web graphs Each page is a vertex
Each edge represent a hyperlink
Directed
GPS: highway maps
Each city is a vertex
Each freeway segment is an undirected edge
Undirected
Data Structure and Algorithm

5. Using Adjacency Matrix Using Adjacency List
Representing Graphs
Using Adjacency Matrix
Using Adjacency List
Data Structure and Algorithm

6. Adjacency Matrix Representation
Example: A 1 2 3 4 1 2 4 3 a d b c
An adjacency matrix represents the graph as a n x n matrix A:
A[i, j] = 1 if edge (i, j)  E (or weight of edge)
= 0 if edge (i, j)  E
Space Complexity: O(V2)
Data Structure and Algorithm

7. Adjacency List Representation1 2 3 4 2 3 1 2 4 3 a d b c 3 3
Space Complexity: O(V+E)
Data Structure and Algorithm