Prove that a simple graph is connected, Mathematics

Assignment Help:

Prove that a simple graph is connected if and only if it has a spanning tree.   

Ans: First assume that a simple graph G has a spanning  tree T.  T consists of every node of G.  By the definition of a tree, there is a path among any two nodes of T.  As T is a subgraph of G, there is a path among each pair of nodes in G. Hence G is connected.   

Here now let G is connected. If G is a tree then nothing to prove. If G is not a tree, it must consist of a simple circuit. Let G has n nodes. We can choose (n - 1) arcs from G in such type of a way that they not form a circuit. It results into a subgraph comprising all nodes and only (n - 1) arcs. So by definition this subgraph is a spanning tree.


Related Discussions:- Prove that a simple graph is connected

Classical probability, Classical Probability Consider the experiment o...

Classical Probability Consider the experiment of tossing a single coin. Two outcomes are possible, viz. obtaining a head or obtaining a tail. The probability that it is a tail

Statistics, If a mean score is 89 with a standard deviation of 8 points. Wh...

If a mean score is 89 with a standard deviation of 8 points. What is the least score you can make and be in the top 20%?

Create a circular table with no restrictions, 1. Four different written dri...

1. Four different written driving tests are administered by a city. One of these tests is selected at random for each applicant for a drivers license. If a group of 2 women and 4 m

What is metric system explain in brief, What is Metric System explain in br...

What is Metric System explain in brief? There are two main systems of measurement in use today. Most countries in the world use the Metric System. In only three countries-the

On income and return from shares, a company declares a semu annual dividend...

a company declares a semu annual dividend on 5%.a man has 400 shares of the company.if his annual income from the share is rs 1000 find the face value of each share?

Definite integral, Definite Integral : Given a function f ( x ) which is c...

Definite Integral : Given a function f ( x ) which is continuous on the interval [a,b] we divide the interval in n subintervals of equivalent width, Δx , and from each interval se

Algebra 1, Im having trouble with this word problem: The three Math Idol j...

Im having trouble with this word problem: The three Math Idol judges have been eliminating contestants all day! The number of one-step equations and two-step equations who have be

Deflation, Deflation Indexes may be utilized to deflate time series so...

Deflation Indexes may be utilized to deflate time series so that comparisons among periods may be made in real terms. This is a process of decreases a value measured in cur

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd