Abstract: The graphs of small order are the fundamental structures of Graph Theory. One way to study their structure is to define the relation G < H whenever both G and H are simple graphs on n vertices and G is isomorphic to a subgraph of H. Then all the unlabeled simple graphs on n vertices form a partially ordered set (poset), and the structure of this poset gives us information about all these graphs.
We have recently studied the structure of the posets of graphs of orders 4, 5, 6, 7 and 8. The present talk will report on our latest extension of this work, to the poset of graphs of order 9. It represents a very significant increase in order of magnitude of the data.