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Chapter 11: Problem 5

Find graphs that have the following adjacency matrices. a. \(\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 2 \\ 1 & 2 & 0\end{array}\right] \quad\) b. \(\left[\begin{array}{lll}0 & 2 & 0 \\ 2 & 1 & 0 \\\ 0 & 0 & 1\end{array}\right]\)

Short Answer

Expert verified
For adjacency matrix a: \(\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 1 & 2 & 0 \end{bmatrix}\) The graph has vertices 1, 2, and 3. Vertex 1 is connected to vertex 3, vertex 2 is connected to vertices 2 (itself) and 3 (with two parallel edges), and vertex 3 is connected to vertices 1 and 2 (with two parallel edges). For adjacency matrix b: \(\begin{bmatrix} 0 & 2 & 0 \\ 2 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\) The graph has vertices 1, 2, and 3. Vertex 1 is connected to vertex 2 (with two parallel edges), vertex 2 is connected to vertices 1 (with two parallel edges) and 2 (itself), and vertex 3 is connected to vertex 3 (itself).

Step by step solution

01

Case a: Adjacency Matrix a

First, we need to analyze the adjacency matrix a, which is: \(\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 1 & 2 & 0 \end{bmatrix}\) We can create an adjacency list from this matrix by identifying the values in the matrix that represent edges between vertices: Vertex 1: (1,3) Vertex 2: (2,2), (2,3), (2,3) Vertex 3: (1,3), (2,3), (2,3) From the adjacency list above, the graph would have three vertices, with edges as follows: - Vertex 1: connected to Vertex 3 - Vertex 2: connected to Vertex 2 (itself), Vertex 3 (two parallel edges) - Vertex 3: connected to Vertex 1, Vertex 2 (two parallel edges) Now, draw the graph for the adjacency list above.
02

Case b: Adjacency Matrix b

Now, let's analyze the adjacency matrix b, which is: \(\begin{bmatrix} 0 & 2 & 0 \\ 2 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\) We can create an adjacency list from this matrix by identifying the values in the matrix that represent edges between vertices: Vertex 1: (1,2), (1,2) Vertex 2: (1,2), (1,2), (2,2) Vertex 3: (3,3) From the adjacency list above, the graph would have three vertices, with edges as follows: - Vertex 1: connected to Vertex 2 (with two parallel edges) - Vertex 2: connected to Vertex 1 (with two parallel edges), Vertex 2 (itself) - Vertex 3: connected to Vertex 3 (itself) Now, draw the graph for the adjacency list above.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Adjacency Matrix
An adjacency matrix is a powerful tool used to represent a graph mathematically. It is a square matrix where the number of rows and columns corresponds to the number of vertices in the graph. Each element in the matrix represents the number of edges between the vertices.

Let's break this down:
  • The matrix element at row i and column j, denoted as aij, shows the number of directed edges from vertex i to vertex j. If there is no edge, the entry is zero.
  • If the graph is undirected, the adjacency matrix is symmetric. This means aij will equal aji.
  • For a graph with n vertices, the adjacency matrix is an n x n matrix.
In the provided exercise, two different adjacency matrices are given, each describing a unique graph structure. By examining these matrices, both the direct connections and any existing self-loops, as well as parallel edges, can be easily identified.
Graph Representation
Graphs can be represented in several ways to suit different requirements and applications. Understanding these representations is crucial for efficiently working with graphs.

Some common graph representation methods include:
  • Adjacency Matrix: As described, it is perfect for dense graphs where the majority of possible edges exist.
  • Adjacency List: This is a collection of lists or an array of lists, where each list corresponds to a vertex in the graph and stores the vertices that are adjacent to it. This representation is ideal for sparse graphs, where most of the possible connections are absent, leading to a less memory-intensive solution.
  • Edge List: A list of all the edges in the graph. Each edge is represented as a pair of vertices which it connects. It is very straightforward but less efficient for large graphs with many connections.
In the exercise, converting the adjacency matrix to an adjacency list makes it easier to draw the graph. This change of representation helps to visualize and verify the connections between different vertices quickly.
Vertices and Edges
A graph comprises two fundamental components: vertices and edges.

Here's what each of them represents:
  • Vertices (or Nodes): These are the individual points on a graph. In the context of an adjacency matrix, each row and column corresponds to a vertex.
  • Edges: These are the lines that connect one vertex to another, showing the relationship between them.
    In an undirected graph, edges are bidirectional, meaning they go both ways between any two vertices. In a directed graph, edges have a direction, indicating a one-way relationship.
  • Properties: Self-loops (edges that connect a vertex to itself) and parallel edges (multiple edges connecting the same two vertices) can exist.
    The existence and nature of these edges are easily captured in the adjacency matrix.
In the given problem, understanding vertices and edges helps accurately interpret the graph's connectivity structure from the adjacency matrix, ensuring that relationships between nodes are properly represented.

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Most popular questions from this chapter

If \(G\) is a connected, weighted graph and no two edges of \(G\) have the same weight, does there exist a unique minimum spanning tree for \(G\) ? Justify your answer.

Suppose that in a group of five people \(A, B, C, D\), and \(E\) the following pairs of people are acquainted with each other: \(A\) and \(C, A\) and \(D, B\) and \(C, C\) and \(D, C\) and \(E\). a. Draw a graph to represent this situation. b. Draw a graph that illustrates who among these five people are not acquainted. That is, draw an edge between two people if, and only if, they are not acquainted.

a. Find the complement of the graph \(K_{4}\), the complete graph on four vertices. (See Example 11.1.8.) b. Find the complement of the graph \(K_{3,2}\), the complete bipartite graph on \((3,2)\) vertices. (See Example 11.1.9.)

In each of 8-21, either draw a graph with the given specifications or explain why no such graph exists. Simple graph, connected, six vertices, six edges

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