1. The problem asks to find all simple connected graphs where the distance between every two non-adjacent vertices is exactly 2. 2. Let's analyze the condition: for any two vertices that are not directly connected by an edge, the shortest path between them must be of length 2. 3. This implies the graph has diameter 2, and every pair of non-adjacent vertices shares a common neighbor. 4. Such graphs are known as strongly regular graphs with parameters $(n,k,\lambda,\mu)$ where $\mu=1$ (each pair of non-adjacent vertices has exactly one common neighbor). 5. The simplest examples are complete graphs $K_n$ (distance 1 between all vertices, so not satisfying the condition), so they are excluded. 6. Next, consider cycle graphs $C_n$. For $n=3$, $C_3$ is complete. For $n=4$, $C_4$ has distance 2 between opposite vertices, but the distance between vertices two edges apart is 2, so it fits. 7. For $n>4$, cycles have distances greater than 2 between some vertices, so they do not satisfy the condition. 8. Another example is the complete bipartite graph $K_{n,n}$ with $n\geq 2$. Here, vertices in the same part are not adjacent but have distance 2 via any vertex in the other part. 9. Also, the friendship graph (windmill graph) formed by joining triangles at a common vertex satisfies the condition. 10. In summary, the graphs satisfying the condition are: - Complete bipartite graphs $K_{n,n}$ with $n\geq 2$ - Cycle graph $C_4$ - Friendship graphs (windmill graphs) Final answer: The simple connected graphs where every two non-adjacent vertices are at distance 2 are $C_4$, complete bipartite graphs $K_{n,n}$ for $n\geq 2$, and friendship graphs.