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

How many different words can be formed out from varanasi, Determine how man...

Determine how many different words can be formed out of the letters of the word VARANASI? Ans: 720 different words can be formed out of the letters of the word VARANASI.

Project, elliptical path of celestial bodies

elliptical path of celestial bodies

Commercial maths, if 500kg of food lasts 40 days for 30 men.how many men wi...

if 500kg of food lasts 40 days for 30 men.how many men will consume 675kg of food in 45 days.

Combination, Combination A combination is a group of times whether ord...

Combination A combination is a group of times whether order is not significant. For a combination to hold at any described time it must comprise of the same items however i

Give an example of numerator and denominator, Give an example of Numerator ...

Give an example of Numerator and Denominator? Fractions represent parts of a whole object. Fractions are written using a horizontal line, with one number on top of the line and

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