Graphs and Networking

1
Graphs and Networking

• Tom Roby
• Cal State Hayward

2
Review

• A graph consists of vertices (points) and edges
  (connections)
• The degree of a vertex is the number of edges
  incident to the vertex
• Two vertices are adjacent if there is an edge
  connecting them

3
Adjacency Matrix

• The entries in the adjacency matrix of a graph G
  are
• aij 1 if there is an edge between
  vertices i and j
• 0 otherwise

4
Example

• Find the adjacency matrix for the graph

5
Going backwards

• Find the graph corresponding to the adjacency
  matrix

C4
6
Try it yourself
C4
Note that the same graph can have several
adjacency matrices
7
How to multiply matrices

8
Powers of A

• If A is the adjacency matrix of a graph G, then
  An describes how many ways there are to get from
  vertex i to vertex j in n steps.

A2
A
9
Number of paths of length 2
A2
There are 3 paths of length 2 from node 1 to
node 1
There are 2 paths of length 2 from node 3 to node
3
There is 1 path of length 2 from node 2 to node 3
10
The shortest path problem

• You want to send some information in a network
  from point A to point B
• The cost of sending information depends on the
  length of the path from A to B
• You want to minimize the cost, i.e. find the
  shortest path from A to B

11
Examples of Networks

• Scanned picture Rosen p.491

12
Weighted graphs

• Definition
• A weighted graph is a graph that has a number
  (weight, length, cost, etc.) assigned to each
  edge.
• //pic of a weighted graph