The sum of the degrees of all the vertices in an undirected graph can be calculated by counting the number of edges incident to each vertex. Each edge contributes 2 to the sum of the degrees since it is counted once for each vertex it is incident to.

Let's break down the problem step by step:

1. Counting the degrees:
- Each vertex in the graph has a degree, which represents the number of edges incident to that vertex.
- The sum of the degrees of all the vertices can be denoted as Σ(deg(v)), where deg(v) represents the degree of vertex v.

2. Relationship between degrees and edges:
- In an undirected graph, each edge connects two vertices.
- Therefore, each edge contributes 1 to the degree of each of its incident vertices.
- Since there are n edges in the graph, each edge contributes 2 to the sum of the degrees.

3. Sum of the degrees:
- The sum of the degrees can be calculated as follows:
Σ(deg(v)) = 2n
- This is because each edge contributes 2 to the sum of the degrees, and there are n edges in total.

Now, we need to find the sum of the degrees for an undirected graph with m vertices and n edges.

4. Substituting the values:
- Since the graph has m vertices, the sum of the degrees can be expressed as Σ(deg(v)) = 2n = 2m.
- This means that the sum of the degrees of all the vertices is equal to 2m.

Therefore, the correct answer is option 'C': 2m.