/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 52 Describe the row of an incidence... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 52

Describe the row of an incidence matrix of a graph corresponding to an isolated vertex.

Short Answer

Expert verified
The row will contain only zeroes.

Step by step solution

01

Understanding an Incidence Matrix

An incidence matrix is a matrix used to represent a graph. Each row corresponds to a vertex and each column corresponds to an edge. The entries of the matrix indicate the relationship between the vertices and the edges.
02

Defining an Isolated Vertex

An isolated vertex in a graph is a vertex that has no edges connected to it. This means it does not participate in any edges.
03

Identifying the Row for an Isolated Vertex

Since an isolated vertex has no edges connected to it, its entire row in the incidence matrix will consist of zeroes. There will be no '1's indicating any connection to an edge.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Isolated Vertex
An isolated vertex is a simple yet important concept in graph theory. It refers to a vertex in a graph that is not connected to any other vertex by an edge. This means the vertex stands alone, with no edges beginning or ending at it.

In visual representations, you'll see the isolated vertex as a single point with no lines (edges) touching it. Imagine a network where one computer is not connected to any others; it's isolated from the network.
  • Isolated vertices can appear in various types of graphs, both directed and undirected.
  • They play a crucial role in understanding the structure and properties of the graph.
Graph Representation
Graphs are mathematical structures used to represent pairwise relationships between objects. The objects are called vertices or nodes, and the relationships are called edges. There are several ways to represent a graph, and one common method is using a matrix.

One popular form of graph representation is known as the incidence matrix. This type of matrix helps in illustrating the connections between vertices and edges. In an incidence matrix:
  • Each row represents a vertex.
  • Each column represents an edge.

For example, if a graph has 3 vertices and 2 edges, its incidence matrix will have 3 rows and 2 columns. The matrix entries (discussed next) help describe the exact relationships.
Matrix Entries
Matrix entries in an incidence matrix tell us how vertices and edges are connected. Each entry can either be a 0 or a 1.

Here鈥檚 what these values mean:
  • A '1' in a cell means that the vertex corresponding to that row is connected to the edge corresponding to that column.
  • A '0' means there is no connection between the vertex and the edge.

For an isolated vertex, every entry in its row in the incidence matrix will be 0. This is because there are no edges connecting it to any other vertices.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the edge chromatic number of \(K_{n}\) when \(n\) is a positive integer.

Suppose that a connected planar simple graph with \(e\) edges and \(v\) vertices contains no simple circuits of length 4 or less. Show that \(e \leq(5 / 3) v-(10 / 3)\) if \(v \geq 4 .\)

Suppose that a connected planar graph has six vertices, each of degree four. Into how many regions is the plane divided by a planar representation of this graph?

How many nonisomorphic connected simple graphs are there with \(n\) vertices when \(n\) is \(\begin{array}{llll}{\text { a) } 2 ?} & {\text { b) } 3 ?} & {\text { c) } 4 ?} & {\text { d) } 5 ?}\end{array}\)

Show that \(g(5)=1 .\) That is, show that all pentagons can be guarded using one point. [Hint: Show that there are either \(0,1,\) or 2 vertices with an interior angle greater than 180 degrees and that in each case, one guard suffices.]
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.