Math Problem Statement

Draw the graph that has the given adjacency matrix as its adjacency graph. Label the vertices a, b, c. How many paths of length 3 exist between a and c? Between b and c?

Solution

Part (i): Drawing the Graph

The adjacency matrix provided is:

\begin{bmatrix} 1 & 1 & 2 \\ 0 & 0 & 1 \\ 2 & 1 & 0 \end{bmatrix}$$ This matrix represents a directed graph with three vertices $$a$$, $$b$$, and $$c$$, where: - The element $$A[i][j]$$ indicates the number of directed edges from vertex $$i$$ to vertex $$j$$. Here is the explanation for constructing the graph: 1. **Vertex $$a$$ (row 1):** - $$A[1][1] = 1$$: There is a self-loop at $$a$$. - $$A[1][2] = 1$$: There is one edge from $$a$$ to $$b$$. - $$A[1][3] = 2$$: There are two edges from $$a$$ to $$c$$. 2. **Vertex $$b$$ (row 2):** - $$A[2][1] = 0$$: No edge from $$b$$ to $$a$$. - $$A[2][2] = 0$$: No self-loop at $$b$$. - $$A[2][3] = 1$$: There is one edge from $$b$$ to $$c$$. 3. **Vertex $$c$$ (row 3):** - $$A[3][1] = 2$$: There are two edges from $$c$$ to $$a$$. - $$A[3][2] = 1$$: There is one edge from $$c$$ to $$b$$. - $$A[3][3] = 0$$: No self-loop at $$c$$. I can create a visual graph based on this matrix. --- ### Part (ii): Paths of Length 3 The number of paths of length 3 between vertices $$i$$ and $$j$$ in a graph can be computed using the cube of the adjacency matrix, $$A^3$$. The element $$(A^3)_{ij}$$ gives the number of paths of length 3 from vertex $$i$$ to vertex $$j$$. Let me compute $$A^3$$ to find the answers for: 1. Number of paths of length 3 between $$a$$ and $$c$$. 2. Number of paths of length 3 between $$b$$ and $$c$$. Let me calculate this now.import numpy as np # Define the adjacency matrix A = np.array([ [1, 1, 2], [0, 0, 1], [2, 1, 0] ]) # Compute the cube of the adjacency matrix A_cubed = np.linalg.matrix_power(A, 3) # Extract paths of length 3 for specific vertex pairs paths_a_c = A_cubed[0, 2] # From 'a' to 'c' paths_b_c = A_cubed[1, 2] # From 'b' to 'c' A_cubed, paths_a_c, paths_b_c(array([[11, 8, 13], [ 2, 2, 5], [12, 7, 6]]), 13, 5)(array([[11, 8, 13], [ 2, 2, 5], [12, 7, 6]]), 13, 5)success### Results for Part (ii): 1. The **cube of the adjacency matrix $$A^3$$** is: $$A^3 = \begin{bmatrix} 11 & 8 & 13 \\ 2 & 2 & 5 \\ 12 & 7 & 6 \end{bmatrix}$$ - Each element $$(A^3)_{ij}$$ represents the number of paths of length 3 from vertex $$i$$ to vertex $$j$$. 2. The number of paths of length 3: - From $$a$$ to $$c$$: $$13$$ paths. - From $$b$$ to $$c$$: $$5$$ paths. --- Would you like a detailed explanation of how these calculations work, or assistance with plotting the graph? Here are some related questions to consider: 1. What does the adjacency matrix represent in terms of graph structure? 2. How does matrix multiplication relate to finding paths in a graph? 3. What are self-loops and how are they represented in a graph? 4. Can adjacency matrices be used for undirected graphs, and how would they differ? 5. What other graph properties can be determined using adjacency matrices? **Tip:** Higher powers of adjacency matrices can provide insights into longer paths and connectivity within the graph.

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Math Problem Analysis

Mathematical Concepts

Graph Theory
Adjacency Matrix
Paths in Directed Graphs
Matrix Exponentiation

Formulas

Cube of adjacency matrix A^3 gives the number of paths of length 3 between vertices.
(A^3)_{ij} represents the number of paths of length 3 from vertex i to vertex j.

Theorems

Matrix Multiplication for Path Counting in Graphs

Suitable Grade Level

Undergraduate

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