1. The problem asks for an example of a simple graph with 12 vertices and 35 edges. 2. A simple graph is an undirected graph with no loops or multiple edges between the same pair of vertices. 3. The maximum number of edges in a simple graph with $n$ vertices is given by the formula $$\frac{n(n-1)}{2}$$. 4. For $n=12$, the maximum number of edges is $$\frac{12 \times 11}{2} = 66$$. 5. Since 35 is less than 66, it is possible to have a simple graph with 12 vertices and 35 edges. 6. One example is to take 12 vertices and connect them with edges until you have 35 edges. 7. For instance, you can start with a complete graph on 8 vertices, which has $$\frac{8 \times 7}{2} = 28$$ edges. 8. Then add 4 more vertices and connect each of them to 7 vertices among the original 8, adding $$4 \times 7 = 28$$ edges, but this would exceed 35. 9. Instead, connect the 4 new vertices with fewer edges, for example, connect each new vertex to 2 or 3 vertices from the original 8 to reach a total of 35 edges. 10. The exact drawing is not possible here, but this explanation shows how to construct such a graph. Final answer: A simple graph with 12 vertices and 35 edges exists because 35 is less than the maximum 66 edges possible for 12 vertices.