## Graph theory in Sage

Even though there are already fine guides in the web to start using the Sage software for graph theory, such as this, a video, and this, I wanted to write a simple one for my students. We have been studying the cubic graphs with eight vertices. And we already know the answer, but we want to ask the computer which of those graphs are vertex transitive.

We obtain the first graph putting together two copies of $K_4$:

```sage: k4=graphs.CompleteGraph(4)
sage: g1=k4+k4
sage: g1.is_vertex_transitive()
True

```

We can get a picture of the graph `g1`:

```sage: g1.show()

```

gives:

Now, the cube graph is easy, it is already defined in the Sage library of graphs:

```sage: g2=graphs.CubeGraph(3)
sage: g2.is_vertex_transitive()
True

```

In this case it is nicer to draw the graph in 3d!

```sage: g2.show3d()

```

which should result in something like:

Now for the third graph, we use “LCF notation”. Basically, we note that from vertex 0, we have a cord to the second vertex, then from 1 again to the second vertex. Then the cord from the next vertex (number 2) goes two steps backward, and also the next. This recipe repeats for the next four vertices, hence:

```sage: g3=graphs.LCFGraph(8, [2,2,-2,-2], 2)
sage: g3.is_vertex_transitive()
False

```

For the fourth, we use again the LCF notation:

```sage: g4=graphs.LCFGraph(8, [4,-2,4,2], 2)
sage: g4.is_vertex_transitive()
False

```

For the fifth, we use another strategy. We will add edges to a cycle of length 8:

```sage: g5=graphs.CycleGraph(8)
sage: g5.is_vertex_transitive()
False

```

The last graph is an example of a “circulant graph”: 8 vertices with “steps” of 1 and 4 vertices:

```sage: g6=graphs.CirculantGraph(8,[1,4])
sage: g6.is_vertex_transitive()
True

```