Graphs

Basic properties 1

Applications of Graph Theory

• • • • Car navigation systems Databases Build a bot to retrieve info from Internet Representing computer networks and streams of information 2

Intuitive Notion of Graph

• A graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called

vertices

, and the links that connect some pairs of vertices are called

edges

.

3

Intuitive Notion of Graph

• • A graph is a bunch of vertices (or nodes) represented by circles which are connected by edges, represented by line segments In other words, graphs can be considered as relations on their vertices set 4

Definition of Graph

• • A graph (an undirected or simple graph) G = (V,E ) is a nonempty finite set V (a set of vertices or nodes) together with a set E of edges , where each edge is a subset of V with cardinality 2 (an unordered pair).

A simple graph is bidirectional (undirected) and has no loops (no “self-communication”).

5

Example

1 {1,2} {1,3} 3 2 {2,3} {3,4} {2,4} 4 • • V={1,2,3,4} E={(1,2),(1,3),(2,3),(2,4),(3,4)} 6

1

Example

{1,2} 2 {2,3} {1,3} {3,4} 3 {2,4} 4 • • • This graph may represent a computer network Vertices are labeled to associate with particular computers Each edge can be viewed as the set of its two endpoints 7

Edges

• For a set V with n elements, how many possible edges there?

• This is the number of pairs in V - the number of 2-element subsets of V:

C n

2  

n

!

2!(

n

 2)!

  8

The number of graphs

• How many possible graphs are there for the same set of vertices V ?

• The number of subsets in the set of possible edges. There are n · (n -1) / 2 possible edges, therefore the number of graphs on V is 2 n(n -1)/2 9

Adjacent Vertices

• Vertices are adjacent if they are the endpoints of the same edge. This edge joins the adjacent vertices.

e

1 1 2

e

2

e

3

e

4 3 4 Q: Which vertices are adjacent to 1? How about adjacent to 2, 3, and 4?

10

Adjacent Vertices

1

e

2

e

1 3

e

3 2

e

4 4 1 is adjacent to 2 and 3 2 is adjacent to 1, 3, and 4 3 is adjacent to 1 and 2 4 is adjacent to 2 5 is not adjacent to any vertex 5 11

Incident Vertices and Edges

• A vertex is incident with an edge (and the edge is incident the edge.

with the vertex) if it is the endpoint of

e

1 1 2 5

e

2

e

3

e

4 3 4 • Which edges are incident to 1? How about incident to 2, 3, 4, and 5?

12

Incident Vertices and Edges

e

1 1 2 5

e

2

e

3

e

4 3 4 1 is incident with e 1

, e

2

e

1

, e

2 are incident with 1

e

1

, e

3

, e

4 are incident with 2 2 is incident with e 1

, e

3

, e

4 3 is incident with e 2

, e

3 4 is incident with e 4 5 is not incident with any edge 13

Degree of a Vertex

• The number of edges incident with a vertex is called the degree of this vertex: deg(A) is the degree of A.

e

1 1 2 5

e

2

e

3

e

4 3 4 • • deg(1)=2; deg(2)=3; deg(3)=2; deg(4)=1; deg(5)=0 Theorem . In a graph, the sum of degrees of the vertices equals twice the number of edges 14

Complete Graph

A simple graph is complete if every pair of distinct vertices share an edge. The notation

K n

denotes the complete graph on n vertices.

K

1

K

2

K

3

K

4

K

5 15

Adjacency Matrix

• • • For a digraph G = (V,E ) define a binary matrix

A G

by: Rows, Columns –one for each vertex in V Value at i th row and j th column is  1 if i th vertex connects to j th vertex (i  j )  0 otherwise 16

Adjacency Matrix - Example

e

1 1 2

e

2

e

3

e

4 3 4

A G

       0 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0       17

Adjacency Matrix

• Theorem . The sum of the entries in row

i

of the adjacency matrix of a graph is the degree of the

i

th vertex.

e

1 1 2

A G

       0 1 1 0 1 0 1 1 1 1 0 0

e

2

e

3

e

4 0 1 0 0        deg(1) 3  2; deg(2)  3; deg(3)  4 18

Graph Isomorphism

• • A graph

G

1 is isomorphic to a graph

G

2 , when there is a one-to-one correspondence

f

between the vertices of

G

1 and

G

2 vertices A and B are adjacent in G 1 such that if and only if the vertices

f

(

A

) and

f

(

B

) are adjacent in

G

2 .

The function

f

with

G

2 .

is called an isomorphism of

G

1 19

Graph Isomorphism

 20

Graph Isomorphism Invariant

• A property is said to be a graph isomorphism  invariant if, whenever

G

1 and

G

2 are isomorphic graphs and

G

1 has this property, then so does

G

2 . The properties are: has n vertices  has e edges  has a vertex of degree k 21

Homework

• • Read Section 4.1

Problems (Exercises 4.1) 1, 3, 5, 7 22