1. 9.1 Introduction to Graphs
TABLE 1: Graph Terminology
Type
Edges
Multiple Edges Allowed?
Loops Allowed?
Simple graph
Undirected No
Multigraph
Yes
Pseudograph
Simple directed graph
Directed
Directed multigraph
Mixed Graph
Directed and Undirected

2. Graph Models
Graphs are used extensively for modeling real-world phenomena
Example:
Vertices are states of the US. Edge between v and w if v shares a border with w.

3. More Examples “Friend” graph for Facebook connections
Computer networks
Cities and highway routes
Cities and airline flights
Round robin tournaments, “defeated”
Statements in a program, “must be executed before”
Telephone calls made in a network (“Call” graph)

4. Precedence Graphs Elaborated
S1 a:=0
S2 b:=1
S3 c:=a+1
S4 d:=b+a
S5 e:=d+1
S6 e:=c+d

5. 9.2 Graph Terminology Terms Adjacent: Incident: Degree: Isolated:
Pendant:

6. Example:
A is adjacent to
The degree of vertex b is
The pendant vertices of the graph are

7. The Handshaking Theorem
Let G be a pseudograph with vertex set V and edge set E. Suppose there are e edges, i.e. |E| = e. Then
In other words, “the sum of the degrees of the vertices is twice the number of edges.”

8. Corollary: An undirected graph has an even number of vertices of odd degree.

9. Directed Graph Terminology
Initial vertex, terminal vertex
Indegree
Outdegree

10. Example:
deg − 𝑎 =
deg + 𝑎 =
deg − 𝑏 =
deg + 𝑏 =
deg − 𝑐 =
deg + 𝑐 =
deg − 𝑑 =
deg + 𝑑 =
deg − 𝑒 =
deg + 𝑒 =
deg − 𝑔 =
deg + 𝑔 =

11. “Handshaking Theorem” for Directed Graphs
Let G be a directed multigraph with vertex set V and edge set E. Suppose there are e edges, i.e. |E| = e. Then
In other words, “the sum of the indegrees of the vertices is equal to the sum of the outdegrees of the vertices, and both are equal to the number of edges.”

12. The “Underlying Undirected Graph”

13. Complete Graphs
A simple undirected graph is a complete graph if all possible edges are present.

14. Other Specialized Graphs
The Cycle Cn
The Wheel Wn
The n-Cube Qn

15. Bipartite Graphs
A graph G = (V,E) is said to be bipartite if and only if there is a two set partition {V1, V2} of V such that every edge e in E has one of its endpoints in V1 and the other in V2.

16. The Complete Bipartite Graph Kmn
𝐾 2,3
𝐾 2,1

17. Bipartite Graphs
Theorem: A simple graph is bipartite if and only if it is possible to assign one of two different colors to each vertex of the graph so that no two adjacent vertices are assigned the same color

18. Subgraphs, Unions, Intersections, and Complements
A graph G = (V′, E′) is a subgraph of graph H= (V,E) if and only if V′  V and E′  E.
The union of two graphs is formed using the union of the two vertex sets and the union of the two edge sets.
The intersection Is defined similarly.
The complement of a graph is the graph which is formed using the same vertex set as the original graph , but which has an edge between two vertices if and only if the original graph does not have such an edge.

19. Examples: