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

What is the prime factorization of 84, What is the prime factorization of 8...

What is the prime factorization of 84? This is the only answer choice which has only PRIME numbers. A prime number is a number along with two and only two distinct factors. In

Unipolar and bipolar boolean inputs, A 4-input Neuron has weights (1,-1,  0...

A 4-input Neuron has weights (1,-1,  0,  0.5.Calculate the network output when the following input vectors are applied. For calculation assume: a. f(net) = unipolar bina

Three person problem of points, Three-person Problem of Points: Pascal, Fer...

Three-person Problem of Points: Pascal, Fermat and their old friend the Chevalier de Mere each put $10.00 into a pot, and agree to play a game that has rounds. Each player has the

Solve sin (3t ) = 2 trig function, Solve sin (3t ) = 2 . Solution T...

Solve sin (3t ) = 2 . Solution This example is designed to remind you of certain properties about sine and cosine.  Recall that -1 ≤ sin (θ ) ≤ 1 and -1 ≤ cos(θ ) ≤ 1 .  Th

matlab, how to solve simplex method using matlab

how to solve simplex method using matlab?

Word problems, Ana has hiked 4 1/2 miles. She is 2/3 of the way along the t...

Ana has hiked 4 1/2 miles. She is 2/3 of the way along the trail. How long is the trail?

Law of Cosines, The law of cosines can only be applied to acute triangles. ...

The law of cosines can only be applied to acute triangles. Is this true or false?

How do children learn maths?, HOW DO CHILDREN LEARN? : Have you ever tried...

HOW DO CHILDREN LEARN? : Have you ever tried teaching a young child what "ball" means? Did you do it by a lot of verbal description" Or did you let the child actually handle a b

Detemine multiplying a polynomial by a monomial, Detemine Multiplying a Pol...

Detemine Multiplying a Polynomial by a Monomial? To multiply a polynomial by a monomial, use the distributive property. Let's start by talking about ordinary numbers. Say th

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