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

Solving geometry using algebra, if one side of a square is increased 4 inch...

if one side of a square is increased 4 inches and an adjacement side is multiplied by 4, the perimeter of the resulting rectangle is 3 times the perimeter of the square. find the s

ALGEBRA, FIND PRODUCT (-41)*(102)

FIND PRODUCT (-41)*(102)

Permutation, A telephone dialled is numbered 0to9. if 0is dialled first the...

A telephone dialled is numbered 0to9. if 0is dialled first the caller is connected to the international exchange system.find the number of local calls that can be rung if a local n

Example of function, Example  Suppose the demand and cost functio...

Example  Suppose the demand and cost functions are given by          Q = 21 - 0.1P and C = 200 + 10Q Where,          Q - Quantity sold

The mode -measures of central tendency, The mode - It is one of the me...

The mode - It is one of the measures of central tendency. The mode is defined as a value in a frequency distribution that has the highest frequency. Occasionally a single valu

Equation of the plane x + 4y 3z = 1, Find the equation of the plane thro...

Find the equation of the plane through (2, 1, 0) and parallel to x + 4y   3z = 1.

Solving equations, darien agrees to sponsor her sister $8 plus $1 for every...

darien agrees to sponsor her sister $8 plus $1 for every mile she walks.Write an expression to show her total money

Completely factored polynomial, Factoring polynomials Factoring polynom...

Factoring polynomials Factoring polynomials is done in pretty much the similar manner.  We determine all of the terms which were multiplied together to obtain the given polynom

Determine the probability - conditional probability, A bag of 28 tulip bulb...

A bag of 28 tulip bulbs contains 12 red tulip bulbs, 9 yellow tulip bulbs, and 7 purple tulip bulbs. Two bulbs are selected without replacement. Determine, a) The probability th

Test, a piece of ribbon measures 2,25 meters . it is cut in half . how long...

a piece of ribbon measures 2,25 meters . it is cut in half . how long is one half of the ribbon

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