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

Testing the difference between two sample means-illustration, An observatio...

An observation was made concerning reading abilities of males and females. The observation leads to a conclusion that females are faster readers than males. The observation was bas

How to make equations of conics easier to read, How to Make Equations of Co...

How to Make Equations of Conics Easier to Read ? If you want to graph a conic sections, first you need to make the equation easy to read. For example, say you have the equatio

Circumference of a circle, How far will a bowling ball goes in one rotation...

How far will a bowling ball goes in one rotation if the ball has a diameter of 10 inches? (π = 3.14) 1. 78.5 in 2. 31.4 in 3. 62.8 in 4. 15.7 in 2. The circumfere

Precalculus help, tsunami equation A sin (b * t) + k what is b supposed t...

tsunami equation A sin (b * t) + k what is b supposed to be if t is time a is amplitude and k is average water level (not exact value of b just what is it)

Numerical analysis, Please,I Want to know and study for stability on predi...

Please,I Want to know and study for stability on predictor -corrector for numerical integration method

What did she pay per pound, Mona purchased one and a half pounds of turkey ...

Mona purchased one and a half pounds of turkey at the deli for $6.90. What did she pay per pound? Divide the cost of the turkey by the weight; $6.90 ÷ 1.5 = $4.60.

Solve-|x2-5x+4/x2-4|

x^2-5x+4 can written in roots as (x-1)*(x-4) x^2-4 can be written interms of (x-2)(x+2).so [(x-1)(x-4)/(x-2)(x+2)]

Area between curves, Area between Curves In this section we will be fi...

Area between Curves In this section we will be finding the area between two curves. There are in fact two cases that we are going to be looking at. In the first case we des

Diffrence between integers and rational numbers, Q. Give basic Diffrence be...

Q. Give basic Diffrence between Integers and Rational Numbers? Ans. Integers The integers are positive and negative whole numbers. The integers are closed under ad

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