Lecture 6 – Induction Examples & Introduction to Graph Theory

You may want to download the the lecture slides that were used for these videos (PDF).

1. Induction Exercises & a Little-O Proof

We start this lecture with an induction problem: show that n2 > 5n + 13 for n ≥ 7. We then show that 5n + 13 = o(n2) with an epsilon-delta proof.  (10:36)

2. Alternative Forms of Induction

There are two alternative forms of induction that we introduce in this lecture. We can argue by contradiction, or we can use strong induction. (1:50)

3. An Introduction to Graph Theory

What is a graph? We begin our journey into graph theory in this video. Graphs are defined formally here as pairs (V, E) of vertices and edges. (6:25)

4. Notation & Terminology

After the joke of the day, we introduce some basic terminology in graph theory. (3:57)

5. First Theorem in Graph Theory

Two times the number of edges is equal to the sum of the degrees in a graph. (4:07)

6. Corollary to the Theorem

The number of vertices of odd degree in any graph must be even.  We see an example of how this result can be applied. (2:41)

7. The Notion of a Subgraph

This video defines a subgraph of a graph. (2:05)

8. Paths in Graphs, Hamiltonian Paths, Size of Paths

Any sequence of n > 1 distinct vertices in a graph is a path if the consecutive vertices in the sequence are adjacent. The concepts of Hamiltonian path, Hamiltonian cycle, and the size of paths are defined. (10:47)

9. Connected Graphs

This video introduces the concepts of connected and disconnected graphs.  It also discusses how a graph may be stored in a file.  (10:07)

10. Cycles & Cliques

This video provides precise definitions of cycles and cliques in graphs. Also defined are loose points.  (8:50)

11. Questions for Thought

Is it easy to tell whether a graph on 2n vertices has a clique of size n? There are many questions we can ask if we are given a graph G=(V,E) with |V| = n. Is G connected? Does G have a path on at least n/2 vertices? We need to distinguish between such yes/no questions that are easy, and yes/no questions that are not, and we need to decide whether we would rather defend a “yes” answer or a “no” answer. (10:20)

12. Isomorphic Graphs

In this video we define isomorphic graphs and discuss whether it is easy to determine when two graphs are isomorphic. (4:01)