1. **State the problem:** Find the degree of vertex C in the given undirected graph. 2. **Recall the definition:** The degree of a vertex in an undirected graph is the number of edges connected to that vertex. 3. **List the neighbors of vertex C:** According to the problem, vertex C is connected to A, B, D, E, and G. 4. **Count the connections:** Vertex C has edges to 5 vertices: A, B, D, E, and G. 5. **Conclusion:** Therefore, the degree of vertex C is $5$. **Final answer:** The degree of vertex C is $5$.