Properties of distinct graphs with given nodes degree

Question

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.

Answer 1

Very different graphs can have the same degree sequence; here is a simple example.

Consider graphs with $12$ vertices, each of degree $2$. Such graphs are unions of cycles. It can have $1,2,3$, or $4$ components. Its clique number can be $2$ or $3$. The independence number can be $6$ (e.g., if the graph is $C_{12}$), $5$ (if the graph is $2$ copies of $C_3$ and one of $C_6$), or $4$ (if the graph is $4$ copies of $C_3$). Its chromatic number can be $2$ (e.g., for $C_{12}$) or $3$ (if it has an odd component). It can be bipartite (if all of its components are even) or not (if it has an odd component).