To determine the minimum value of $k$ in a $k$-regular graph of order 20 (i.e., a graph with 20 vertices where each vertex has degree $k$) such that the distance between any two vertices is at most 2, we need to ensure that every pair of vertices is either directly connected or connected via one intermediary vertex.

Key Points:

1. Regular Graph: Each vertex has exactly $k$ neighbors.
2. Distance Constraint: The distance between any two vertices is at most 2.
3. Order 20: The graph has 20 vertices.

Steps to Solve:

1. Direct Connection: For any two vertices to be at most 2 steps away, each vertex should be directly connected to as many other vertices as possible.
2. Neighbor's Neighbor: If a vertex $v$ is not directly connected to another vertex $u$, then $u$ should be connected to a vertex that is a neighbor of $v$.

To find the minimum $k$, let’s start with an estimation based on the connectivity:

1. Counting Connections:

   • Each vertex is directly connected to $k$ vertices.
   • Each of those $k$ vertices is directly connected to $k-1$ other vertices (excluding the initial vertex).

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