Problem 12 - Discrete Mathematics with Applications

Question

Draw the graph with the given adjacency matrix.

Step-by-Step Solution

Answer

Given the adjacency matrix, identify the connected vertices by examining the non-zero elements in the matrix. Draw vertices as points labeled by their number, and connect the identified connected vertices with lines. If the graph is weighted, label each edge with its weight. Ensure the resulting graph matches the adjacency matrix by checking the connections between vertices.

Step 1: Understand the adjacency matrix

The adjacency matrix is a square matrix, where its rows and columns represent the vertices of the graph. The value at the (i, j) position in the matrix indicates the weight of the edge between vertices i and j. In case of an unweighted graph, the value will be 1 if there is an edge between vertices i and j and 0 otherwise.

Step 2: Identify the connected vertices

To determine which vertices are connected by an edge, examine each element of the adjacency matrix. If the value of the (i, j) element is non-zero, it means there is an edge between vertices i and j. Record these pairs of vertices for the next step.

Step 3: Draw the vertices and edges

Begin by drawing the vertices of the graph. You can represent each vertex by a point, labelled by its number. Next, refer to the list of connected vertices generated in step 2. Draw a line between each pair of connected vertices. In case of a weighted graph, label the edge with its weight.

Step 4: Check the resulting graph

After drawing the graph, compare it with the given adjacency matrix. Ensure that all vertices are properly connected according to the adjacency matrix, by re-checking the non-zero values of the matrix. If everything matches, it's complete. If there are inconsistencies, revise the graph accordingly. Throughout the process, refer to the given adjacency matrix as needed to ensure that the graph accurately reflects the connections specified by the matrix.