Determine whether the following graph is a simple graph or multigraph, and obtain the adjacency matrix.
Understand the Problem
The question is asking to determine if two given graphs are simple graphs or multigraphs and to find their corresponding adjacency matrices.
Answer
Graph a is a simple graph with adjacency matrix $A_a = \begin{bmatrix} 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix}$ and graph b is a multigraph with adjacency matrix $A_b = \begin{bmatrix} 0 & 2 & 2 & 2 & 2 \\ 2 & 0 & 2 & 2 & 2 \\ 2 & 2 & 0 & 2 & 2 \\ 2 & 2 & 2 & 0 & 2 \\ 2 & 2 & 2 & 2 & 0 \end{bmatrix}$.
Answer for screen readers
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Graph a is a simple graph with adjacency matrix: $$ A_a = \begin{bmatrix} 0 & 1 & 1 & 0 \ 1 & 0 & 1 & 1 \ 1 & 1 & 0 & 0 \ 0 & 1 & 0 & 0 \end{bmatrix} $$
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Graph b is a multigraph with adjacency matrix: $$ A_b = \begin{bmatrix} 0 & 2 & 2 & 2 & 2 \ 2 & 0 & 2 & 2 & 2 \ 2 & 2 & 0 & 2 & 2 \ 2 & 2 & 2 & 0 & 2 \ 2 & 2 & 2 & 2 & 0 \end{bmatrix} $$
Steps to Solve
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Identify the Type of Graphs
- A simple graph has no loops or multiple edges between the same set of vertices.
- A multigraph may have multiple edges or loops.
Graph a Analysis:
- In graph a, there are no loops and no multiple edges; therefore, it is a simple graph.
Graph b Analysis:
- In graph b, there are multiple edges between some vertices; therefore, it is a multigraph.
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Determine Adjacency Matrix for Graph a
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Assign the vertices:
- 1: Vertex 1
- 2: Vertex 2
- 3: Vertex 3
- 4: Vertex 4
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The edges present are: (1,2), (1,3), (2,3), (2,4).
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The adjacency matrix is constructed as follows:
- Rows and columns represent the vertices.
- Matrix cell value is 1 if there is an edge, otherwise 0.
The resulting matrix is: $$ A = \begin{bmatrix} 0 & 1 & 1 & 0 \ 1 & 0 & 1 & 1 \ 1 & 1 & 0 & 0 \ 0 & 1 & 0 & 0 \end{bmatrix} $$
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Determine Adjacency Matrix for Graph b
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Similarly, assign vertices:
- 1: Vertex 1
- 2: Vertex 2
- 3: Vertex 3
- 4: Vertex 4
- 5: Vertex 5
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The edges between the vertices in graph b can be assumed to be each vertex connected to every other vertex, including multiple edges between nodes.
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Assuming multiple edges, the adjacency matrix can be represented with higher values where similar edges exist: $$ A = \begin{bmatrix} 0 & 2 & 2 & 2 & 2 \ 2 & 0 & 2 & 2 & 2 \ 2 & 2 & 0 & 2 & 2 \ 2 & 2 & 2 & 0 & 2 \ 2 & 2 & 2 & 2 & 0 \end{bmatrix} $$
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Graph a is a simple graph with adjacency matrix: $$ A_a = \begin{bmatrix} 0 & 1 & 1 & 0 \ 1 & 0 & 1 & 1 \ 1 & 1 & 0 & 0 \ 0 & 1 & 0 & 0 \end{bmatrix} $$
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Graph b is a multigraph with adjacency matrix: $$ A_b = \begin{bmatrix} 0 & 2 & 2 & 2 & 2 \ 2 & 0 & 2 & 2 & 2 \ 2 & 2 & 0 & 2 & 2 \ 2 & 2 & 2 & 0 & 2 \ 2 & 2 & 2 & 2 & 0 \end{bmatrix} $$
More Information
- A simple graph means each pair of vertices is connected by at most one edge. A multigraph allows multiple edges or loops. The adjacency matrix provides an easy way to represent the connections in a graph mathematically.
Tips
- Confusing between simple graphs and multigraphs by not checking for loops or multiple connections.
- Incorrectly assigning edges in the adjacency matrix by overlooking some connections.
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