1. Lecture 10: Graphs Graph Terminology Special Types of Graphs
Representing Graphs
Graph Isomorphism

2. Basic Terminology An edge connects two vertices
Two vertices are adjacent if they are connected
An edge is incident with the two vertices it connects
Vertices are the endpoints of the edge connecting them
The degree of a vertex is the number of incident edges
An isolated vertex has degree zero (0)
A pendant vertex has degree one (1)

3. The Handshaking "Theorem"4 5 7 3 6 2 1

4. The "First Theorem" of Graph Theory
Every undirected graph has an even number of vertices of odd degree. E O

5. A Theorem for Directed Graphs
In a graph with directed edges the in-degree of a vertex v, denoted by deg-(v), is the number of edges with v as their terminal vertex.
The out-degree of v, denoted by deg+(v), is the number of edges with v as their initial vertex.
Let G=(V,E) be a graph with directed edges. Then

6. Complete Graphs Kn
A complete graph is a simple graph with one edge between every pair of vertices. K1 K2 K3 K4 K5 K6
How many edges are there in a complete graph of n vertices?
First we note that each vertex of Kn has degree n-1.
Using the Handshaking Theorem, we have
2e = S deg(v) = n*(n-1),
therefore
e = n*(n-1)/2.

7. Cycles C3 C4 C5 C6

8. Wheels W3 W4 W5 W6

9. The n-Cube Qn Q1 Q2 Q3

10. Q4 - The 4D Hypercube

11. Connection Machine
The Connection Machine was a series of supercomputers that grew out of Danny Hillis's research in the early 1980s at MIT on alternatives to the traditional von Neumann architecture of computation. The Connection Machine was originally intended for applications in artificial intelligence and symbolic processing, but later versions found greater success in the field of computational science.
CM-2
CM-5

12. Some Complete Bipartite Graphs
K2,3
K3,3
K3,5
K2,6
The "first theorem" of planar graph theory - K3,3 is not planar.

13. The Arc Reversal Algorithm
The arc-reversal algorithm has applications in computer communications, parallel processing, flow analysis, scheduling and Bayesian Networks.

24. The Assignment Problem

25. Maximal Matching Problem

26. Subgraph of a Graph

27. Union of Graphs

28. Graph Isomorphisms

29. Adjacency Matrix Graph Representation
For an n-node graph we build an nxn array with 1's indicating edges and 0's no edge the main diagonal of the matrix is unused unless a node has an edge connected to itself. If graph is weighted, 1's are replaced with edge weight values
adjacency matrix
A B C D E F G H A B C D E F G H A D F C H B E G

30. Summary Basic Terminology Some Classic Theorems Types of Graphs
K, C, W, Q
Bipartite and Complete Bipartite Graphs
Graph Problems and Algorithms
Arc Reversal Algorithm
Assignment Problem
Maximal Matching
Graphs and Subgraphs
Graph Isomorphisms
Adjacency Matrix