Introduction:
In this problem, we are given a simple, connected, undirected graph G. We need to add a new node v to G and connect it to all the odd degree vertices of G. We are asked to determine the properties of the resultant graph.

Solution:
To solve this problem, let's consider the properties of the resultant graph after adding node v and connecting it to all the odd degree vertices of G.

1. Connected:
The original graph G is connected, and since we are only adding a single node v and connecting it to the odd degree vertices, the resultant graph will also be connected.

2. Simple:
The original graph G is simple, i.e., it does not contain any self-loops or multiple edges between the same pair of vertices. Adding a new node v and connecting it to the odd degree vertices will not introduce any self-loops or multiple edges. Therefore, the resultant graph will also be simple.

3. Undirected:
The original graph G is undirected, and adding a new node v and connecting it to the odd degree vertices will not change the directionality of any edges. Therefore, the resultant graph will also be undirected.

4. Degree of v:
By construction, node v is connected to all the odd degree vertices of G. Therefore, the degree of v in the resultant graph will be equal to the number of odd degree vertices in G.

5. Degree of other vertices:
Since we only added edges from node v to the odd degree vertices of G, the degree of all other vertices in the resultant graph remains the same as in the original graph G.

6. Eulerian:
A graph is Eulerian if and only if all its vertices have even degree. In the original graph G, some vertices have odd degree. By adding node v and connecting it to all the odd degree vertices, the resultant graph ensures that all vertices have even degree. Therefore, the resultant graph is Eulerian.

Conclusion:
From the above analysis, we can conclude that the resultant graph after adding node v and connecting it to all the odd degree vertices of G is connected, simple, undirected, and Eulerian. Therefore, the correct answer is option 'D' (Eulerian).