Math Problem Statement

Write the adjacency matrix and transition matrix of the following graph.

Solution

To find the adjacency matrix and transition matrix for the given directed graph, let’s go through the following steps.

Step 1: Identify the nodes and their connections

The nodes are labeled as aa, bb, cc, and dd. Based on the directed arrows, we can identify the connections:

  • From aa:

    • a→ba \to b
    • a→ca \to c

  • From bb:

    • b→db \to d

  • From cc:

    • c→ac \to a
    • c→bc \to b
    • c→dc \to d

  • From dd:

    • d→cd \to c

Step 2: Construct the Adjacency Matrix

An adjacency matrix AA is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. Here, we use a matrix where:

  • A[i][j]=1A[i][j] = 1 if there is a directed edge from node ii to node jj,
  • A[i][j]=0A[i][j] = 0 otherwise.

Let’s label the nodes in the order a,b,c,da, b, c, d, and create the matrix:

0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{bmatrix}$$ ### Step 3: Construct the Transition Matrix A transition matrix $$T$$ normalizes the adjacency matrix by ensuring each row sums up to 1 (if a node has outgoing edges). For each element $$T[i][j]$$ in the transition matrix, we divide by the out-degree of the corresponding node $$i$$ (the total number of outgoing edges from node $$i$$). The out-degrees for each node are: - **Node $$a$$:** 2 outgoing edges (to $$b$$ and $$c$$) - **Node $$b$$:** 1 outgoing edge (to $$d$$) - **Node $$c$$:** 3 outgoing edges (to $$a$$, $$b$$, and $$d$$) - **Node $$d$$:** 1 outgoing edge (to $$c$$) Using these out-degrees, we get: $$T = \begin{bmatrix} 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 1 \\ \frac{1}{3} & \frac{1}{3} & 0 & \frac{1}{3} \\ 0 & 0 & 1 & 0 \\ \end{bmatrix}$$ ### Summary - **Adjacency Matrix:** $$A = \begin{bmatrix} 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{bmatrix}$$ - **Transition Matrix:** $$T = \begin{bmatrix} 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 1 \\ \frac{1}{3} & \frac{1}{3} & 0 & \frac{1}{3} \\ 0 & 0 & 1 & 0 \\ \end{bmatrix}$$ Would you like a further breakdown or have any questions? Here are some related questions that might deepen your understanding: 1. How is the adjacency matrix used in graph theory? 2. Why do we need to normalize to create the transition matrix? 3. Can you explain the relationship between transition matrices and Markov chains? 4. How would the matrices change if an additional edge was added to the graph? 5. What would the transition matrix look like for an undirected graph? **Tip:** For directed graphs, always note the direction of arrows when constructing adjacency matrices, as it directly affects the matrix's values.

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Math Problem Analysis

Mathematical Concepts

Graph Theory
Adjacency Matrix
Transition Matrix

Formulas

Adjacency matrix: A[i][j] = 1 if there is a directed edge from node i to node j, 0 otherwise
Transition matrix: T[i][j] = A[i][j] / (sum of row i in A) if row sum is not zero

Theorems

Matrix Representation of Graphs
Markov Chains (related to transition matrix)

Suitable Grade Level

College

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