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

Solve the right triangle, 1. Solve the right triangle. B = 135     c = 3...

1. Solve the right triangle. B = 135     c = 3.72 A  ≈ ____°    (round to the nearest tenth as needed) 2.  Solve the right triangle, where  a =4 and b =10 The length of

What is the probability of getting a royal flush, Q. What is the probabilit...

Q. What is the probability of getting a Royal Flush? Ans. Five cards are picked from a standard deck of 52 cards. How many different hands of five cards are possible? What

Shares and dividents., how much money will required to buy 200,rupees 25 sh...

how much money will required to buy 200,rupees 25 shares at premium of rupees 2

Factor Fiction, Ok this is true or false wit a definition. The GCF of a pai...

Ok this is true or false wit a definition. The GCF of a pair of numbers can never be equal to one of the numbers.

Solve 8 cos2 (1 - x ) + 13 cos(1 - x )- 5 = 0 trig function, Solve 8 cos 2 ...

Solve 8 cos 2 (1 - x ) + 13 cos(1 - x )- 5 = 0 . Solution Now, as specified prior to starting the instance this quadratic does not factor.  Though, that doesn't mean all i

Solve the limit problem, Solve the Limit problem as stated  Limit x tends...

Solve the Limit problem as stated  Limit x tends to 0 [tanx/x]^1/x^2 is ? lim m tends to infinity [cos (x/m)] ^m is? I need the procedure of solving these sums..

Evaluate algebraic word problems, Evaluate algebraic word problems: A ...

Evaluate algebraic word problems: A utility has three nuclear facilities which supply a total of 600 megawatts (Mw) of electricity to a particular area.  The largest facility

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