The graph editor allows you to create graphs on the screen, add or delete vertices and edges, move vertices around, see the adjacency and incidence matrices, colour vertices and edges etc.

From the Windows Start menu, select Programs then Mathematics then MT356 Software and finally Graph Editor. This loads the Graph Editor—click on OK in the opening screen.

To create your own graph using the graph editor:

• from the File menu, select New;
• click on OK in the dialogue box that appears;
• draw the vertices of the graph by clicking the right mouse button at appropriate places in the window. (Each right mouse-button click places a vertex on the screen; to remove a vertex, first select it with the left mouse-button and then press the delete key.)
• draw the edges of the graph by using the right mouse button to drag the pointer (a cross) between the two vertices you wish to join. (To remove an edge, select it with the left mouse button and then press the delete key.)

Once you have created your graph you can:

• change its appearance by moving the vertices around—select a vertex and drag it to a new position;
• change its appearance by colouring the vertices or edges—select a vertex or edge and then click one of the small colour buttons in the toolbar (or select one of the colours from the Colour menu);
• view the adjacency matrix—click on the    button (to hide the adjacency matrix, click again on the    button);
• view the incidence matrix—click on the    button (to hide the incidence matrix, click again on the    button);
• label the vertices—double click on a vertex (or select the vertex and chose Label from the Edit menu), type the required label and click on OK.

Close the existing window, open a new window and create the following graph. Colour red all vertices of degree 2, colour green all vertices of degree 3, and colour blue all vertices of degree 4.

Now create, in the same window a different (that is, non isomorphic) graph with the same number of vertices of each degree as the graph above.

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The Graph Database contains a database of all simple graphs with up to seven vertices—there are 1252 of them. With the graph editor still running, open the Graph Database (Start ® Programs ® Mathematics ® MT356 Software ® Graph Database).

The graphs in the database are listed as G1, G2 ..., G1252. For each graph, the database lists the number of vertices n, the number of edges m and the degree sequence. Clicking on a graph in the list opens a window containing a diagram of the graph.

For example, G100 is the following graph with 6 vertices and 6 edges.

It is possible to select (from those graphs currently listed in the main window) those graphs with a certain number of vertices or edges or having a certain degree sequence.

1. To select all (simple) graphs with 6 vertices and 6 edges:

• click on the     button in the tool bar (or pull down the Select menu and choose No. Vertices (n)) and enter 6 in the dialogue box that appears. This selects all the graphs with 6 vertices; there are 156 of them, listed as G33, ..., G208.
• click on the  

button in the tool bar (or pull down the Select menu and choose No. Edges (m)) and enter 6 in the dialogue box that appears. This selects from the 156 currently listed all those graphs with 6 edges; there are 21 of them, listed as G86, ..., G106.

2. To select all (simple) graphs with degree sequence (1, 2, 2, 3, 4, 4)

• pull down the Select menu and choose All. This restores the entire database of 1252 graphs.
• click on the    button

button in the tool bar (or pull down the Select menu and choose Degree sequence) and enter 1,2,2,3,4,4 in the dialogue box that appears. This selects all the graphs with degree sequence (1, 2, 2, 3, 4, 4); there are just 2 such graphs, G137 and G138.

Diagrams of graphs in the graph database cannot be manipulated directed. However, it is possible to copy a diagram from the database and paste it into the graph editor where it can be manipulated. For example, to paste a copy of G137 into the graph editor:

• make the window containing G137 active by clicking somewhere in it;
• from the Edit menu select Copy;
• switch to the Graph Editor application;
• from File menu select New and click on OK in the resulting dialogue box;
• from the Edit menu select Paste.

Note that you can also paste a copy of G138 into the same Graph Editor window if you wish. To do this repeat the steps above except do not select a new Graph Editor window—just paste G138 into the existing window.

Try now to manipulate the graphs so that you obtain the following picture to compare G137 and G138.

To see why the two graphs are different (not isomorphic), it may be useful to colour vertices of different degrees with different colours (for example, colour red all vertices of degree 2, colour green all vertices of degree 3, etc.)