How can you describe this kind of graph?

Question

I know for sure that this is a simple directed graph. But I cannot say that this is a ring graph/network, because node 3 has a degree of 4. But as I imagine this, you cannot go to node 7 from node 3 if the preceding node is node 2, and you cannot go to node 4 from node 3 if the preceding node is node 6. That means the only way to traverse this graph is to start from one node and then go to the adjacent node that has a number greater than the current node (except for node 7 to node 1). What kind of graph is this? Thanks in advance!

http://img231.imageshack.us/img231/5492/graphl.jpg

Answer 1

Yes, it is a simple directed graph. It is also an Eulerian graph with exactly one Eulerian circuit. That's probably its most interesting property.

Answer 2

You could redraw the graph as a standard Eulerian Graph without the addition rules by dividing node 3 into two nodes, node 3 and node 6A, where 2-3-4 and 6-6A-7 are the only valid paths through the two nodes. At this point the graph would look like a figure eight. The nodes could then be rearranged in a circle while maintaining the topology.