To solve the problem of finding the maximum number of edges in a graph with 12 vertices where no two cycles share a common vertex, we can follow these key observations:

Key Insights:

1. Cycle Properties: A cycle of length (k) (simple graph) has exactly (k) edges and (k) vertices (edges = vertices for a cycle).
2. Vertex-Disjoint Cycles: Since no two cycles share a vertex, all cycles are independent. To maximize edges, we want as many vertices as possible to be part of cycles (because cycles contribute more edges than non-cycle structures like trees).
3. Non-Cycle Vertices: If there are remaining vertices not in cycles, they form a tree (max edges = (r-1) for (r) vertices), which is fewer edges than a cycle of the same size.

Maximum Edges Calculation:

For 12 vertices:

• All 12 vertices can be covered by vertex-disjoint cycles (e.g., 4 triangles, one 12-cycle, etc.).
• Each cycle contributes equal edges to its vertex count, so total edges = total vertices in cycles = 12.

Answer: (\boxed{12})

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