Math notes for Ams 301 course, Lecture notes of Mathematics

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AMS 301

Fred Rispoli

Chapter 1, Sections 1.1 and 1.

Today Course Information Elements of Graph Theory, Chapter 1 Graph Models Isomorphism

A B D^ C A multiple edge A loop A B D^ C No loops or multiple edges are allowed in our discussion (other people allow them, sometimes calling them multigraphs or pseudographs ). Eg. #2: A directed graph (a.k.a. an oriented graph , or a digraph ) is a graph with little arrows on the edges. Eg. #3: Here the order of the letters when you write the edge is important. The directed (or oriented) edge from the vertex A to the vertex B is written. A is the tail of , and B is the head.

AB

→ AB → In-deg(A) = 0 Out-deg(A) = 2

Examples of new graphs people are looking at is the

Facebook graph or social graphs in general.

A set of vertices in a graph which have no edges connecting any of them is called an independent set of vertices. An edge cover of a graph is a set of vertices in the graph so that every edge in the graph is incident with at least one of the vertices in the set. In a digraph, the analogy to an edge cover is called a vertex basis. Here the set of vertices is such that every edge in the graph has one of those vertices as its tail. To be a vertex basis though, the set of vertices must be as small as possible to cover the graph. A B D^ C A B D^ C An independent set {A,D} An edge cover {B,C} A vertex basis is {A,B}

Class Exercise Find the largest set of independent vertices in

the graph below.

a e i b (^) c d f j g h k . Indepdent Vertices Example

10 Matching Example

Suppose we have 4 people and 4 jobs, and that

various people are qualified for various jobs. Use

the graph to help find a one to one matching of

people to jobs.

A

B

C

D

a

b

c

d

Street Surveillance Example Put a cop on selected corners (Vertices) so that every street (edge) is watched. Use as few cops as possible. That is, find a minimal edge cover for the graph below. a e i b c d f j g h k

For each graph below determine the number of edges

and the sum of the degrees.

Number of edges Degree Sum

Handshaking

Conjecture?

Class Exercise

In the beginning of every session of the US Supreme

Court the 9 justices all shake hands. How many

handshakes take place?

Example

What do the following graphs have in common?

A B

C D

G G’

• Two graphs G and G’ are isomorphic if :
  • There exists a one-to-one correspondence between vertices

in G and G’, such that

• There is an edge between a and b in G if and only if there is

an edge between the corresponding vertices and in.

• The definition for oriented graphs is the same, except the head

and tail of each edge of G must correspond to the head and tail

in.

Definition of Isomorphism G 

G 

K n , the complete graphs on n vertices K 2

K 3

K 5

K 8

K 6

K 4

K 1

The complement of a graph

The complement of G has all the edges that are missing in G.

These are the edges that would have to be added to make the

complete graph.

G

K 6

G