Graph Theory

Step 1: Examine the graphs and adjacency matrices.
The adjacency matrix for a graph shows the connections (edges) between the nodes. A 1 represents an edge between two nodes, and a 0 represents no edge.

Step 2: Match each graph with its corresponding adjacency matrix.
- **A**: The graph in option A corresponds to adjacency matrix **I** because it shows two nodes connected to others in the same pattern.

- **B**: The graph in option B corresponds to adjacency matrix **II** because of the pattern of connected nodes.

- **C**: The graph in option C corresponds to adjacency matrix **III** based on its connections.

- **D**: The graph in option D corresponds to adjacency matrix **IV** based on the connections between the nodes.

Step 3: Conclusion.
Thus, the correct matching is **A - I, B - II, C - III, D - IV**.

Step 1: Understand the minimum spanning tree.
A minimum spanning tree (MST) connects all vertices in a graph with the least possible total edge weight, without forming any cycles.

Step 2: Evaluate the given edges and their weights.
The edges and their corresponding weights are:

- (1, 2) with weight 11

- (3, 6) with weight 14

- (4, 6) with weight 21

- (2, 6) with weight 24

- (1, 4) with weight 31

- (3, 5) with weight 36

To form a minimum spanning tree, we select the edges with the smallest weights first, ensuring there are no cycles.

Step 3: Conclusion.
The correct graph will be the one that uses the minimum weight edges, such as (1, 2), (3, 6), and (4, 6), and avoids creating cycles. This corresponds to option (1).