Math Problem Statement

Consider a weighted graph G with 7 vertices {A, B, C, D, E, F, G}, which is represented by the following adjacency matrix.

Consider a weighted graph G with 7 vertices {A, B, C, D, E, F, G}, which is represented by the following adjacency matrix.

Solution

To solve the question, we need to perform Prim's algorithm on the given graph, starting from vertex AA. The adjacency matrix defines the weights between vertices. Prim's algorithm builds a Minimum Cost Spanning Tree (MCST) by starting with a single vertex and successively adding the smallest-weight edge that connects a new vertex.

Adjacency Matrix (Vertices A,B,C,D,E,F,GA, B, C, D, E, F, G):

The matrix you provided represents the graph. We'll decode this matrix to determine edge weights between vertices:

  1. The matrix rows and columns represent vertices in order A,B,C,D,E,F,GA, B, C, D, E, F, G.
  2. An entry in the ithi^{th} row and jthj^{th} column represents the weight of the edge between vertex ii and jj. A value of 0 indicates no edge exists.

Steps of Prim's Algorithm:

  1. Start at vertex AA. Add AA to the tree.
  2. Find the smallest-weight edge that connects a vertex in the tree to a vertex not in the tree. Add the corresponding vertex.
  3. Repeat until all vertices are included.

Execution:

Initialization:

  • Start with AA.
  • Add the smallest-weight edge connected to AA.

Iterative Steps:

We'll follow these steps by inspecting the matrix and selecting edges in ascending order of weight:

  1. Step 1 (Start with AA):

    • Edges from AA: AC(6)A \to C (6), AF(9)A \to F (9), AG(7)A \to G (7).
    • Smallest weight is 66 for ACA \to C.
    • Add CC to the tree: {A,C}\{ A, C \}.
  2. Step 2:

    • Edges from {A,C}\{ A, C \}: AF(9)A \to F (9), AG(7)A \to G (7), CG(8)C \to G (8).
    • Smallest weight is 77 for AGA \to G.
    • Add GG to the tree: {A,C,G}\{ A, C, G \}.
  3. Step 3:

    • Edges from {A,C,G}\{ A, C, G \}: AF(9)A \to F (9), CG(8)C \to G (8), GF(2)G \to F (2).
    • Smallest weight is 22 for GFG \to F.
    • Add FF to the tree: {A,C,G,F}\{ A, C, G, F \}.
  4. Step 4:

    • Edges from {A,C,G,F}\{ A, C, G, F \}: AF(9)A \to F (9), CG(8)C \to G (8), FE(5)F \to E (5).
    • Smallest weight is 55 for FEF \to E.
    • Add EE to the tree: {A,C,G,F,E}\{ A, C, G, F, E \}.
  5. Step 5:

    • Edges from {A,C,G,F,E}\{ A, C, G, F, E \}: FD(1)F \to D (1), CG(8)C \to G (8).
    • Smallest weight is 11 for FDF \to D.
    • Add DD to the tree: {A,C,G,F,E,D}\{ A, C, G, F, E, D \}.
  6. Step 6:

    • Remaining vertex is BB. Edge DB(8)D \to B (8).
    • Add BB to the tree.

Final Order:

The vertices are added in the following order: A,C,G,F,E,D,BA, C, G, F, E, D, B.

Answer:

The correct option is: A,C,G,F,E,D,BA, C, G, F, E, D, B.


Would you like me to verify or expand on any specific step? Let me know if you have questions!


Related Questions:

  1. How does Prim's algorithm differ from Kruskal's algorithm?
  2. What is the significance of a Minimum Cost Spanning Tree?
  3. Can Prim's algorithm be implemented using a priority queue? How?
  4. What is the time complexity of Prim's algorithm for dense graphs?
  5. How does the choice of the starting vertex in Prim's algorithm affect the outcome?

Tip:

When solving graph problems, sketching the graph based on the adjacency matrix can help visualize connections and edge weights better.

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Math Problem Analysis

Mathematical Concepts

Graph Theory
Minimum Cost Spanning Tree (MCST)
Prim's Algorithm

Formulas

Edge weight comparison to find the minimum edge

Theorems

Prim's algorithm for MCST

Suitable Grade Level

Undergraduate level (Graph Algorithms)