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Chapter 3: Problem 57

Draw a digraph that has the given adjacency matrix. $$\left[\begin{array}{llll} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 \end{array}\right]$$

Short Answer

Expert verified
Draw vertices 1, 2, 3, 4 and add edges 1鈫�2, 2鈫�1, 2鈫�4, 3鈫�2, 4鈫�1, 4鈫�3, 4鈫�4.

Step by step solution

01

Understand the Adjacency Matrix

The given adjacency matrix represents a directed graph with 4 vertices. In this matrix, rows represent the start vertex and columns represent the end vertex. A value of 1 indicates a directed edge from the starting vertex to the ending vertex, while a value of 0 indicates no direct edge between those vertices.
02

Identify Edges from Matrix Row 1

The first row is \([0, 1, 0, 0]\). Here, vertex 1 has a directed edge to vertex 2, as indicated by the '1' in the second column. There are no edges from vertex 1 to vertices 3 or 4.
03

Identify Edges from Matrix Row 2

The second row is \([1, 0, 0, 1]\). This means vertex 2 has directed edges to vertices 1 and 4. The '1' in the first column indicates an edge from vertex 2 to vertex 1, and the '1' in the fourth column indicates an edge from vertex 2 to vertex 4.
04

Identify Edges from Matrix Row 3

The third row is \([0, 1, 0, 0]\). Vertex 3 has a directed edge to vertex 2. This is shown by the '1' in the second column, indicating an edge from vertex 3 to vertex 2.
05

Identify Edges from Matrix Row 4

The fourth row is \([1, 0, 1, 1]\). Vertex 4 has directed edges to vertices 1, 3, and 4 itself. The '1's in the first, third, and fourth columns indicate edges from vertex 4 to vertices 1, 3, and 4, respectively.
06

Draw the Digraph

Using the identified edges, draw a digraph with vertices 1, 2, 3, and 4. Add directed edges based on the matrix: \[\trellis_{(1,2)}\] (from 1 to 2), \[\trellis_{(2,1)}\] (from 2 to 1), \[\trellis_{(2,4)}\] (from 2 to 4), \[\trellis_{(3,2)}\] (from 3 to 2), \[\trellis_{(4,1)}\] (from 4 to 1), \[\trellis_{(4,3)}\] (from 4 to 3), and \[\trellis_{(4,4)}\] (from 4 to 4 itself).

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Key Concepts

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

Directed Graph
When we talk about a directed graph, or digraph, we refer to a set of vertices connected by edges, where the edges have a direction. This means each edge leads from one vertex to another specified vertex. In a directed graph:
  • Each edge has a start (or source) and an endpoint.
  • The direction is usually indicated by an arrow.
  • It is possible for two vertices to have edges going in both directions (not necessarily forming a cycle).
  • Self-loops, where a vertex is connected to itself, are allowed.
Directed graphs are very useful in depicting real-world situations where relationships are not necessarily mutual, like in prerequisite courses, hierarchy structures, or web page links.
Digraph
A digraph is simply the short form of a directed graph. Every digraph is made of vertices and directed edges. While the structure may resemble that of an undirected graph, the distinguishing factor is that digraphs have directed edges.
  • In mathematical terms, a digraph is denoted as a pair \( G = (V, E) \), where \( V \) is a set of vertices and \( E \) is a set of edges.
  • Each edge is an ordered pair \( (u, v) \) indicating that there is a directed edge from \( u \) to \( v \).
Digraphs are extensively used in computer science and mathematics to model pairwise relations between objects.
Graph Theory
Graph theory is a field of mathematics that studies the properties and applications of graphs. A graph, in this context, is a collection of nodes connected by edges.
  • These graphs can be used to solve problems related to connectivity, flow, scheduling, and more.
  • Graph theory provides tools to analyze the structure of networks and predict their behavior.
Graph theory not only addresses the study of directed graphs but also explores many other types of graphs in network theory, including undirected graphs, weighted graphs, and many more.
Matrix Representation
Matrix representation is a method used to describe a graph using a matrix. The most common types of matrix representations in graph theory are adjacency matrices and incidence matrices.
  • An adjacency matrix is a square matrix used to represent a finite graph. The elements indicate whether a pair of vertices are adjacent or not.
  • For digraphs, the matrix is not necessarily symmetric, since the direction of the edge is important.
  • A value of 1 signifies the presence of an edge, while 0 signifies no edge.
This representation is efficient for computer algorithms and allows easy computation of graph properties such as the degree of each vertex, the presence of cycles, and transitive closures.

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Most popular questions from this chapter

Draw a digraph that has the given adjacency matrix. $$\left[\begin{array}{lllll} 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 \end{array}\right]$$

Give a counterexample to show that the given transformation is not a linear transformation. $$T\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{l} x+1 \\ y-1 \end{array}\right]$$

Draw a graph that has the given adjacency matrix. $$\left[\begin{array}{llll} 0 & 1 & 1 & 1 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right]$$

(a) Show that the product of two \(2 \times 2\) stochastic matrices is also a stochastic matrix. (b) Prove that the product of two \(n \times n\) stochastic matrices is also a stochastic matrix. (c) If a \(2 \times 2\) stochastic matrix \(P\) is invertible, prove that \(P^{-1}\) is also a stochastic matrix.

The three types of elementary matrices give rise to five types of \(2 \times 2\) matrices with one of the following forms: \\[ \begin{array}{c} {\left[\begin{array}{cc} k & 0 \\ 0 & 1 \end{array}\right] \text { or }\left[\begin{array}{cc} 1 & 0 \\ 0 & k \end{array}\right]} \\ {\left[\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right]} \\ {\left[\begin{array}{cc} 1 & k \\ 0 & 1 \end{array}\right] \text { or }\left[\begin{array}{cc} 1 & 0 \\ k & 1 \end{array}\right]} \end{array} \\] Each of these elementary matrices corresponds to a linear transformation from \(\mathbb{R}^{2}\) to \(\mathbb{R}^{2}\). Draw pictures to illustrate the effect of each one on the unit square with vertices at \((0,0),(1,0),(0,1),\) and (1,1)
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