/*! 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 6 Find all (loop-free) nonisomorph... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 6

Find all (loop-free) nonisomorphic undirected graphs with four vertices. How many of these graphs are connected?

Short Answer

Expert verified
There are 11 non-isomorphic, loop-free, undirected graphs with four vertices. Out of these, 6 are connected.

Step by step solution

01

Derive all possible undirected graphs with four vertices

Start with a vertex and systematically add edges to create all possibilities. With four vertices, A, B, C, D, start with no edges (i.e., four isolated vertices), then add one, two, three, up to six edges (which is the full complement for four vertices in an undirected graph without loops).
02

Identify Non-Isomorphic Graphs

After deriving all possible graphs in step 1, identify those that are non-isomorphic. An isomorphism is a graph that appears different but is in fact the same because the vertices can be relabeled to make one graph identical to another. If there's no such relabeling possible, then two graphs are non-isomorphic. In this case, with four vertices, the following 11 graphs are non-isomorphic: No edges, 1 edge, 2 edges in a line, 2 edges forming a 'V', 3 edges in a line, 3 edges forming a 'U', 4 edges forming a 'square', 4 edges forming 'T', 5 edges and all 6 edges (forming a complete graph).
03

Determine the connected graphs

From the graphs identified in the previous step, find out which are connected. A graph is connected if there's a path between any two vertices. The connected graphs among our 11 non-isomorphic cases are: 2 edges in a line, 3 edges in a line, 4 edges forming a 'square', 4 edges forming 'T', 5 edges and all 6 edges (forming a complete graph). So, there are six connected graphs out of the 11 non-isomorphic graphs.

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!

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

a) For \(n \geq 3\), how many different Hamilton cycles are there in the complete graph \(K_{n}\) ? b) How many edge-disjoint Hamilton cycles are there in \(K_{21} ?\) c) Nineteen students in a nursery school play a game each day where they hold hands to form a circle. For how many days can they do this with no student holding hands with the same playmate twice?

Let \(n \in \mathbf{Z}^{+}\)with \(n \geq 4\). How many subgraphs of \(K_{n}\) are isomorphic to the complete bipartite graph \(K_{1,3}\) ?

a) Find the number of edges in \(Q_{8}\). b) Find the maximum distance between pairs of vertices in \(Q_{8}\). Give an example of one such pair that achieves this distance. c) Find the length of a longest path in \(Q_{8}\).

a) How many subgraphs \(H=(V, E)\) of \(K_{6}\) satisfy \(|V|=\) \(3 ?\) (If two subgraphs are isomorphic but have different vertex sets, consider them distinct.) b) How many subgraphs \(H=(V, E)\) of \(K_{6}\) satisfy \(|V|=4 ?\) c) How many subgraphs does \(K_{6}\) have? d) For \(n \geq 3\), how many subgraphs does \(K_{n}\) have?

Prove that for \(n \geq 2\), the hypercube \(Q_{n}\) has a Hamilton cycle.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics 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.