Consider a weighted complete graph G on the vertex set {v1, v2, ..vn} such…

Answer: 2n — 2

Reasoning:

• The edge weight is 2|i − j|, so the smallest possible edge weight is 2, occurring exactly for edges between consecutive vertices v1–v2, v2–v3, …, v_{n−1}–v_n.

• There are n − 1 such consecutive edges and they form a simple path that connects all vertices, so these edges by themselves form a spanning tree.

• By Kruskal's algorithm (or by choosing the smallest available edges), the algorithm will pick these n − 1 edges of weight 2 and stop, giving total weight (n − 1) × 2 = 2n − 2.

• As a lower bound, any spanning tree has n − 1 edges and each edge has weight at least 2, so the total weight cannot be less than 2(n − 1). This proves optimality.