Hello smithhallHere's a graph that satisfies the conditions to get you started. The vertices are , with degree as follows:

(maximum)

(pendant)

The sum of the degrees of the vertices is 38, because there are 19 edges, each edge connecting 2 vertices.

To construct a second graph that's not isomorphic to this one, you could re-draw this one by removing the edge and replacing it by the edge . This will produce the following specification:

(maximum)

(pendant)

I suspect there aren't many other possibilities.

Grandad