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

What is the area of the square in simplified form, If the side of a square ...

If the side of a square can be expressed as a2b 3 , what is the area of the square in simplified form? Since the formula for the area of a square is A = s 2 , then by substitut

Algebraic models, Establish appropriate algebraic models for each of the fo...

Establish appropriate algebraic models for each of the following sets of data. You can use technology to assist. Plot them on grids and demonstrate how you have established each mo

Integral calculus, how to change order and variable in multiple integral

how to change order and variable in multiple integral

Find where the breakdown occurred and his original speed, A cyclist, after ...

A cyclist, after riding a certain distance, stopped for half an hour to repair his bicycle, after which he completes the whole journey of 30km at half speed in 5 hours.  If the bre

Evaluate the measure of the larger angle, Two angles are complementary. The...

Two angles are complementary. The calculate of one angle is four times the measure of the other. Evaluate the measure of the larger angle. a. 36° b. 72° c. 144° d. 18°

Matrices, suppose you a business owner and selling cloth. the following rep...

suppose you a business owner and selling cloth. the following represents the number of items sold and the cost for each item. use matrix operation to determine the total revenue ov

Average cost function, Average cost function : Now let's turn our attentio...

Average cost function : Now let's turn our attention to the average cost function. If C ( x ) is the cost function for some of the  item then the average cost function is,

Regression coefficient, 4x+3y+7=0 and 3x+4y+8=0 find the regression coeffic...

4x+3y+7=0 and 3x+4y+8=0 find the regression coefficient between bxy and byx.

Arithmetic/geometric sequences and binomial expansion, find s10 for the ari...

find s10 for the arithmetic sequenxe inwhich a1=5 and a10=68

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