Discussion
Consider the undirected weighted graph G with 3 vertices, whose adjacency matrix is given as :
\(G = \begin{bmatrix} 0 & 2 & 2 \\ 2 & 0 & 2 \\ 2 & 2 & 0 \end{bmatrix}\)
Which of the following is true ?
Answer: C
Graph G has 3 distinct minimum spanning trees, each of cost 4.
The matrix describes a triangle on three vertices with every edge of weight 2. A spanning tree on three vertices uses 2 of the 3 edges, and there are 3 ways to leave one edge out. Each tree costs 2 + 2 = 4, so all three are minimum spanning trees.
A connected graph always has a minimum spanning tree, the MST is unique only when ties do not allow alternatives, and the spanning trees here all have the same cost (and option D speaks of a graph H that is not given).
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