Problem 16 - Discrete Mathematics with Applications

Question

Find the adjacency matrices of the graphs in Exercises \(1-6.\) Draw the graph with the given list representation.

Step-by-Step Solution

Answer

For the graph with list representation \( G: \{ a\rightarrow b, b\rightarrow c, c\rightarrow d, d\rightarrow a \} \), we first drawn the graph with four nodes and directed edges as described in the list. Next, we created a \(4 \times 4\) adjacency matrix with 0's in its diagonal entries. Then, we filled the entries of the matrix with 1 if there is a directed edge connecting the nodes, otherwise with 0. The adjacency matrix for this graph is: \[ A = \begin{bmatrix} 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{bmatrix} \] Perform these steps for each of the remaining graphs in the exercise.

Step 1: Extract the list representation of the graph

We will choose one of the six graphs and extract its list representation. For this example, let's choose a graph with the given list representation: \[ G: \{ a\rightarrow b, b\rightarrow c, c\rightarrow d, d\rightarrow a \} \]

Step 2: Draw the graph based on the list representation

According to the given list representation, there are four nodes and four directed edges. The nodes of the graph are \(a\), \(b\), \(c\), and \(d\). Now let's draw the graph. 1. Put four nodes on the paper labeled as \(a\), \(b\), \(c\), and \(d\). 2. Draw an arrow (directed edge) from node \(a\) to node \(b\). 3. Draw an arrow from node \(b\) to node \(c\). 4. Draw an arrow from node \(c\) to node \(d\). 5. Draw an arrow from node \(d\) to node \(a\). Now the graph is drawn.

Step 3: Create the adjacency matrix based on the graph

An adjacency matrix is a square matrix used to represent a finite graph. To find the adjacency matrix for our graph, follow these steps: 1. Make a \(4 \times 4\) matrix, where the number of rows and columns is equal to the number of nodes in the graph because there are four nodes (\(a\), \(b\), \(c\), and \(d\)). 2. Fill in the diagonal entries with 0 since the nodes do not have loops. 3. Fill in the remaining entries of the matrix as follows: - If node \(i\) is connected to node \(j\) by a directed edge (arrow), put a 1 in the \((i, j)\) entry in the matrix. Otherwise, put a 0. 4. The adjacency matrix for our graph will be: \[ A = \begin{bmatrix} 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{bmatrix} \] Now we have found the adjacency matrix for the given graph. Follow these steps for the remaining five graphs in the exercise.

Key Concepts

Graph Theory

Graph theory is a field of mathematics and computer science that deals with problems involving a set of points connected by lines. These points are typically called vertices (or nodes), and the lines are edges (or links). In our daily lives, we encounter diverse examples of graph theory, like social networks where users are vertices and friendships are edges.

Understanding graph theory is vital for various applications including computer network design, analysis of molecular structures in chemistry, and solving puzzles like the Königsberg bridge problem, which laid the foundations of the field. Consider the subway system of a city: stations serve as vertices, while the tracks are the edges that connect them. This kind of structural modeling is essential for efficient planning and problem-solving.

Directed Graphs

Directed graphs, also known as digraphs, are graphs where the edges have a direction associated with them, indicated by an arrow. Therefore, the connection between two nodes is not bidirectional by default, unlike in undirected graphs. In our exercise, the graph drawn from a given list representation is directed. This means that if node 'a' points to node 'b', it doesn't necessarily imply that 'b' points back to 'a'.

Applications of Directed Graphs

Directed graphs are highly significant in representing systems where the relationship between entities is not symmetric, such as one-way streets in a city map, food chains in ecology, or the flow of computation in a computer program.

Matrix Representation of Graphs

Matrix representation of graphs is an essential concept in graph theory, providing a convenient way to study and analyze graphs mathematically. One common type of matrix representation is the adjacency matrix, highlighted in our exercise. The adjacency matrix for a graph with 'n' vertices is an 'n x n' matrix where the entry at row 'i' and column 'j' represents the presence or absence of an edge from vertex 'i' to vertex 'j'. In the given example of a directed graph, the adjacency matrix is used to encode the graph's structure compactly and unambiguously.

As indicated in the exercise, the adjacency matrix's main diagonal is filled with zeros because loops, where a vertex points to itself, are not present. This representation makes it easier to apply mathematical concepts and algorithms to the study of graphs, such as determining the shortest path or exploring connectivity between nodes.