# Graphs of lines on del Pezzo surfaces

Let $k$ be an algebraically closed field. To any del Pezzo surface $S$ over $k$ we may associate its graph of lines, which has one vertex for each line and an edge (with multiplicity if required) between two vertices if and only if the corresponding lines insect. For me a "line" means a $(-1)$-curve. It is well-known that the graph only depends on the degree of the del Pezzo surface, so we obtain a series of graphs $G_d$ for $d \leq 8$.

For $d \geq 3$ these graphs are all well-studied by graph theorists for their own sake, for example they have their own wikipedia pages. We have

However this is where my knowledge stops.

Are the graphs $G_1$ and $G_2$ similarly well-studied? e.g. Do they have special names or special properties which uniquely identify them?

I appreciate that this question is slightly vague, but I hope the reader understands what I am after.

For example all the above mentioned graphs are implemented in sage. Are $G_1$ and $G_2$ also implemented in sage, or perhaps another computer algebra package? Alternatively, is there a way to construct $G_2$ and $G_1$ from the $E_7$ and $E_8$ root systems? If necessary I am willing to consider the corresponding graphs obtained by removing the multiple edges.

• Yes, these are the vertex-edge graphs of the $E_7$ and $E_8$ polytopes. This is written up in Section 8.2.5 (in the published version) of Dolgachev, Classical Algebraic Geometry.
– user5117
Jun 24, 2014 at 14:00
• Thanks, I think this is exactly what I was looking for! If you would like to post this as an answer, I will accept it. Jun 24, 2014 at 14:42
• Glad it was helpful. Comment now answerified!
– user5117
Jun 24, 2014 at 14:51

## 1 Answer

(Making my comment into an answer...)

Yes, these are the vertex-edge graphs of the $E_7$ and $E_8$ polytopes. This is written up in Section 8.2.5 (in the published version) of Dolgachev, Classical Algebraic Geometry.