/*! 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 50 How many subgraphs with at least... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 50

How many subgraphs with at least one vertex does \(K_{2}\) have?

Short Answer

Expert verified
3

Step by step solution

01

Define the Graph

A complete graph with 2 vertices is denoted as \(K_2\). It consists of 2 vertices and 1 edge connecting them.
02

Identify Possible Subgraphs

List all possible subgraphs that can be formed with at least one vertex from \(K_2\). These include: one vertex alone (there are two such subgraphs, one for each vertex) and the whole graph itself, which includes both vertices and the edge.
03

Count the Subgraphs

From Step 2, we have identified 3 subgraphs: one subgraph for each of the vertices and one subgraph that includes both vertices and the edge. Hence, the total number of subgraphs with at least one vertex is 3.

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.

Complete Graph
A complete graph, often denoted as \(K_n\), is a type of graph in graph theory where every pair of distinct vertices is connected by a unique edge. This means if you have \(n\) vertices, there will be \[\frac{n(n-1)}{2}\] edges in the graph. For instance, in the graph \(K_2\), there are 2 vertices and 1 edge connecting them.
Subgraphs
A subgraph is a smaller part of a graph that consists of a subset of its vertices and edges. The subgraphs of a given graph can have varying numbers of vertices and edges, but they must all be parts of the original graph. For instance, in the case of \(K_2\), the possible subgraphs with at least one vertex include:
  • A subgraph containing only one of the two vertices (there are two such subgraphs)
  • The entire graph itself, which includes both vertices and the edge connecting them
Graph Theory
Graph theory is a field of mathematics that studies the properties and applications of graphs. Graphs are structures made up of vertices (also called nodes) connected by edges. This theory is useful for modeling relationships and solving problems in various fields such as computer science, biology, and social sciences. By understanding concepts like complete graphs and subgraphs, one can analyze and solve complex problems more efficiently.

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

Show that the property that a graph is bipartite is an isomorphic invariant.

Show that a simple graph \(G\) with \(n\) vertices is connected if it has more than \((n-1)(n-2) / 2\) edges.

How many edges does a graph have if its degree sequence is 5, 2, 2, 2, 2, 1? Draw such a graph.

Suppose there is an integer k such that every man on a desert island is willing to marry exactly k of the women on the island and every woman on the island is willing to marry exactly k of the men. Also, suppose that a man is willing to marry a woman if and only if she is willing to marry him. Show that it is possible to match the men and women on the island so that everyone is matched with someone that they are willing to marry.

What do the in-degree and the out-degree of a vertex in a directed graph modeling a round-robin tournament represent?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Applied Mathematics

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.