Consider a graph G = (V, E), where V = {v1, v2, ..., v10}, E={(vi, vj) ∣ 1≤ i ≤ j≤ 10) and weight of each edge (vi , vj) is ∣i – j∣. Find the minimum spanning tree of G and its weight.
Question
Consider a graph G = (V, E), where V = {v1, v2, ..., v10}, E={(vi, vj) ∣ 1≤ i ≤ j≤ 10) and weight of each edge (vi , vj) is ∣i – j∣. Find the minimum spanning tree of G and its weight.
Solution
The graph G described is a complete graph with 10 vertices. The weight of each edge is the absolute difference of the indices of the vertices it connects.
A minimum spanning tree (MST) of a graph is a subset of the edges of the graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight.
In this case, the minimum spanning tree of G can be constructed by connecting each vertex vi to the vertex vi+1, for i from 1 to 9. This is because the weight of the edge (vi, vi+1) is 1, which is the smallest possible weight for any edge in the graph.
The edges of the MST are then: {(v1, v2), (v2, v3), (v3, v4), (v4, v5), (v5, v6), (v6, v7), (v7, v8), (v8, v9), (v9, v10)}.
The total weight of the MST is the sum of the weights of these edges. Since each edge has weight 1, the total weight is 9.
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