# Directed Graphs on at most 9 vertices with some properties

Let $\mathcal{F}$ be the family of strongly connected digraphs with $\leq 9$ vertices, up to isomorphism.

Asuume that a directed edge from vertex $v$ to $w$, if exists, is unique; and in this case, there is no directed edge from $w$ to $v$.

Is this family easy to enumerate, say using GAP or SAGE?

Note: The elements of $\mathcal{F}$ have following properties:

(1) Each graph $G$ in $\mathcal{F}$ is connected and has $\leq 9$ vertices.

(2) For any distinct vertices $v,w$ of $G$, there is a directed path from $v$ to $w$.

(3) [Edit-from Smylic's comment]Every vertex has at least one incoming and one outgoing edge.

(4) If there is directed edge from $v$ to $w$, then no directed edge from $w$ to $v$.

Is it easy to obtain enumeration of the family $\mathcal{F}$, say using some softwares such as GAP, SAGE etc.?

I have handeled GAP for groups, but no idea whether we can handle directed graphs there.

Any suggestions for this problems? Also I would appreciate for suggestions of references on such graphs, their properties.

• Can you give some details on which types of properties you might be interested in? Right now, your question is very broad ... Commented Mar 2, 2017 at 12:58
• I edited and wrote as precise as possible the question; hope that it is clear. Commented Mar 2, 2017 at 13:04
• Did you mean in (3) that vertex has at least one incoming edge and at least one outgoing edge? (Otherwise you are considering only directed cycles.) Commented Mar 2, 2017 at 14:02
• yes, you are right. I edited it. Thanks for noticing it. Commented Mar 2, 2017 at 14:35

N = 5