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

Proof for absolute convergence - sequences and series, Proof for Absolute C...

Proof for Absolute Convergence Very first notice that |a n | is either a n or it is - a n depending upon its sign.  The meaning of this is that we can then say, 0 a n +

What are the average total repair costs per month, An automobile manufactur...

An automobile manufacturer needs to build a data warehouse to store and analyze data about repairs of vehicles. Among other information, the date of repair, properties of the vehic

Lorie, A bourbon that is 51 proof is 25.5% alcohol by volume while one that...

A bourbon that is 51 proof is 25.5% alcohol by volume while one that is 82 proof is 41% alcohol. How many liters of 51 proof bourbon must be mixed with 1.0 liter of 82 proof bourbo

Differential equation, Suppose a fluid (say, water) occupies a domain D? R^...

Suppose a fluid (say, water) occupies a domain D? R^(3 ) and has velocity field V=V(x, t). A substance (say, a day) is suspended into the fluid and will be transported by the fluid

Permatuation and combination problem, How may six digit numbers can be made...

How may six digit numbers can be made in which the sum of the digits is even? Ans = 9*10*10*10*10*5

What is median number of tries it took these participante, The operator of ...

The operator of an amusement park game remain track of how many tries it took participants to win the game. The subsequent is the data from the ?rst ten people: 2, 6, 3, 4, 6, 2, 8

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?

#t, show that a*0=a

show that a*0=a

Graph f(x) = ex and g(x) = e- x - common graph, Graph f ( x ) = e x and g ...

Graph f ( x ) = e x and g ( x ) = e - x . Solution There actually isn't a lot to this problem other than ensuring that both of these exponentials are graphed somewhere.

Sum, As1212uestion #Minimum 100 words accepted#

As1212uestion #Minimum 100 words accepted#

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