Suppose $N$ nodes with degree $v_1,v_2,...,v_n$ are given. How many distinct graphs can be built?
Have all these graphs the same property? I mean they all have the same shortest TSP, They are all planner or non-planner, They all have the same diagonal and any other property of a graph.
In the aggregate I would like to know, When there is a list of nodes degree, Is there any difference in terms of graph properties among all the graphs that can be generated using the nodes degree?
To be not too broad, what properties are the same among the graphs?
Thanks in advance.