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

Stat, defination of uper boundarie .

defination of uper boundarie .

Abstract algebra, How many homomorphism are there from z2 to z3. Zn is grou...

How many homomorphism are there from z2 to z3. Zn is group modulo n

Need some clarity?, THE % PARTICIPATION Feature in a major medical expense ...

THE % PARTICIPATION Feature in a major medical expense policy is 75% with a $100 deductible. how much of a $2,000 bill is the insured responsible for paying?

The mean value theorem, The Mean Value Theorem : In this section we will ...

The Mean Value Theorem : In this section we will discuss the Mean Value Theorem.  Before we going through the Mean Value Theorem we have to cover the following theorem. Ro

Standard normal distribution, Q. Describe Standard Normal Distribution? ...

Q. Describe Standard Normal Distribution? Ans. The Standard Normal Distribution has a mean of 0 and a standard deviation of 1. The letter Z is often used to refer to a sta

How to dealing with exponents on negative bases, How to Dealing With Expone...

How to Dealing With Exponents on Negative Bases ? Exponents work just the same way on negative bases as they do on positive ones: (-2)0 = 1 Any number (except 0) raised to the

Evaluate the rational exponents, Evaluate each of the following.  (a) 2...

Evaluate each of the following.  (a) 25 1/2  (b) 32 1/5 Solution  (a) 25 1/2 Thus, here is what we are asking in this problem.                             2

How to adding polynomials, How to Adding Polynomials? The numerical par...

How to Adding Polynomials? The numerical part of a monomial is called the coefficient. For example, the coefficient of 5x is 5. The coefficient of -7a 2 b 3 is -7. Like

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