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

Probability, Mike sells on the average 15 newspapers per week (Monday – Fri...

Mike sells on the average 15 newspapers per week (Monday – Friday). Find the probability that 2.1 In a given week he will sell all the newspapers [7] 2.2 In a given day he will sel

Fraction, maria has a slice of pizza that is 1/6 of the pizaa.Ben has a sli...

maria has a slice of pizza that is 1/6 of the pizaa.Ben has a slice of pizza that is 1/3 of the pizza, marias slice is bigger.draw pizzas to show how this is possible.

Learning, my math skills are keeping me from getting my ged need help in al...

my math skills are keeping me from getting my ged need help in all areas

Percentage, 7 is what percent of 105?.

7 is what percent of 105?.

Math, how do you do algebra in 4th grade

how do you do algebra in 4th grade

Integers, hi i would like to ask you what is the answer for [-9]=[=5] grade...

hi i would like to ask you what is the answer for [-9]=[=5] grade 7

Compute simple addition, John was doing his homework on vertical addition, ...

John was doing his homework on vertical addition, and had to compute : 5 3+ 3 4  and 6 8 +45 He did the first one easily, just the way his teacher had taught him. He first ad

Probability, julie has 3 hats and 5 scarves. How many ways can she wear a h...

julie has 3 hats and 5 scarves. How many ways can she wear a hat and a scarf?

Determine the value of the unknown side of a right triangle, Determine the ...

Determine the value of the unknown side of a right triangle: The two legs of a right triangle are 5 ft and 12 ft.  How long is the hypotenuse? Now Let the hypotenuse be c ft.

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