Building Graphs with Specific Properties Using Sage
I would like to use Sage to build graphs with particular properties. I know how to call a program called "nauty" to ask Sage to generate, for example, all graphs on 8 vertices with 16 edges with clique number 4. However, I would like to add more properties.
Is it possible to have Sage generate all graphs with clique number 4 so that all vertices that are contained in a 4-clique satisfy a minimum degree condition? Or is it in general possible to have a particular subset of vertices satisfy a degree condition?
Thank you.
Tip: make your question more specific: say exactly what "minimum degree condition" you mean.
Answering a specific question is easier. Extending from that example might then be easy.