1. Chapter 9: Graphs Basic Concepts
Mark Allen Weiss: Data Structures and Algorithm Analysis in Java
Chapter 9: Graphs
Basic Concepts
Lydia Sinapova, Simpson College

2. Graphs – Basic Concepts
Basic definitions: vertices and edges
More definitions: paths, simple paths, cycles, loops
Connected and disconnected graphs
Spanning trees
Complete graphs
Weighted graphs and networks
Graph representation
Adjacency matrix
Adjacency lists

3. Basic Graph Definitions
A graph is a mathematical object that is used to model different situations – objects and processes:
Linked list
Tree (partial instance of graph)
Flowchart of a program
City map
Electric circuits
Course curriculum

4. Vertices and Edges
Definition: A graph is a collection (nonempty set) of vertices and edges
Vertices: can have names and properties
Edges: connect two vertices,
can be labeled,
can be directed
Adjacent vertices: there is an edge between them

5. Example Vertices: A,B,C,D Edges: AB, AC, BC, CD A C B D A B C D Graph1
Two ways to draw the same graph

6. Directed and undirected graphsB C D
These are two different graphs

7. More definitions : Path
A list of vertices in which successive vertices are connected by edges
A B C
B A C D
A B C A B C A B C D
B A B A C A B C D

8. More definitions : Simple Path
No vertex is repeated.
A B C D
D C A
D C B
A B
A B C A B C D

9. More definitions : Cycle
Simple path with distinct edges, except that the first vertex is equal to the last A B C D
A B C A
B A C B
C B A C
A graph without cycles is called acyclic graph.

10. More definitions : Loop
An edge that connects the vertex with itself A B C D

11. Connected and Disconnected graphs
Connected graph: There is a path between each two vertices
Disconnected graph : There are at least two vertices not connected by a path.
Examples of disconnected graphs: A B C D

12. Graphs and Trees
Tree: an undirected graph with no cycles, and a node chosen to be the root A B C E D
Source graph:

13. Graphs and Trees A C E B D C A E B D
Tree1: root A
Tree2: root C

14. A spanning tree of an undirected graph
A sub-graph that contains all the vertices, and no cycles. If we add any edge to the spanning tree, it forms a cycle, and the tree becomes a graph A B C D A B
graph
spanning tree C D

15. Examples A B C D
All spanning trees of the graph on the previous slide

16. Complete graphs
Graphs with all edges present – each vertex is connected to all other vertices A B C D E
Dense graphs: relatively few of the possible edges are missing
Sparse graphs: relatively few of the possible edges are present
A complete graph

17. Weighted graphs and Networks
Weighted graphs – weights are assigned to each edge (e.g. road map)
Networks: directed weighted graphs (some theories allow networks to be undirected) B 2 1 C A 2 4 3 D

18. Graph Representation
Adjacency matrix
Adjacency lists

19. Adjacency matrix – undirected graphs
Vertices: A,B,C,D
Edges: AC, AB, AD, BD
A B C D
  A B C D
The matrix is symmetrical A B C D

20. Adjacency matrix – directed graphs
Vertices: A,B,C,D
Edges: AC, AB, BD, DA
A B C D  A B C D A B C D

21. Adjacency lists – undirected graphs
Vertices: A,B,C,D
Edges: AC, AB, AD, BD
Heads lists
A B C D
B A D
C A
D A B A B C D

22. Adjacency lists – directed graphs
Vertices: A,B,C,D
Edges: AC, AB, BD, DA
Heads lists
A B C
B D
C =
D A A B C D