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Adjacency Matrices 1

Ultrabook / Netbook Software Downloads » Education Softwares » Adjacency Matrices 1

Date Added:

May 26, 2012 02:23 AM

Publisher's Description:

The adjacency matrix of a graph on n vertices is an n x n matrix A = (ai,j) in which the entry ai,j =1 if there is an edge from vertex i to vertex j and is 0 if there is no edge from vertex i to vertex j. A matrix with only zeros and ones as entries is called a (0,1) matrix, pronounced "zero-one matrix".

Java Web Start Activity:
The Java Web Start application below will help you to understand adjacency matrices. Draw a graph in the left pane in the same way as you have done in previous applications. In the right pane you will see the adjacency matrix for the graph you draw. The application uses letters for labels so that they are not confused with the entries in the adjacency matrix.
Java Web Start Application 8.1: Adjacency matrices

Petersen Activity:
To use the Petersen program to see the adjacency matrix of a graph, you should first draw the graph in the program and then click Properties | Adjacency Matrix.

Fig 8.1. The icosahedron and its adjacency matrix

Questions:
In the Petersen program, look at the adjacency matrices of the null graphs N3, N10, N15. Describe in your own words the adjacency matrix of a null graph.
Look at the adjacency matrices of the complete graphs K3, K10, K15. Describe in your own words the adjacency matrix of a complete graph.
Look at the adjacency matrices of a few more graphs. Give an interpretation for the sum of the entries in row i of an adjacency matrix.
Suppose you are told that the adjacency matrix for a simple graph has 5 rows and 5 columns. Suppose you are also told that each row contains three ones and two zeros, why is this impossible?
Investigate the effect of reordering the vertices of a graph on the adjacency matrix.
First get the graph of the cube by selecting Graph | Named Graph | Platonic Graph | Cube | Non Planar.
Make the program use letters for the labels by selecting Picture | Labels | Letters.
Get the adjacency matrix for a cube by selecting Properties | Adjacency Matrix. The result should look like figure 8.2.