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

Dropped down the rational expression to lowest terms, Carry out the indicat...

Carry out the indicated operation and dropped down the answer to lowest terms.  (x 2 - 5x -14/ x 2 -3x+2) .   (x 2 - 4)/x 2 -14x+49) Solution This is a multiplication.

Counters and registers, design a synchronous, recycling, MOD-12 counter wit...

design a synchronous, recycling, MOD-12 counter with D FF''s. Use the states 0000 through 1011 in the counter.

Operation research, approximate the following problem as a mixed integer pr...

approximate the following problem as a mixed integer program. maximize z=e-x1+x1+(x2+1)2 subject to x12+x2 =0

Applications of derivatives rate change, Application of rate change Bri...

Application of rate change Brief set of examples concentrating on the rate of change application of derivatives is given in this section.  Example    Find out all the point

Confidence interval, Confidence Interval The interval estimate or a 'co...

Confidence Interval The interval estimate or a 'confidence interval' consists of a range as an upper confidence limit and lower confidence limit whether we are confident that a

Linear equation, The ratio between the length and breadth of a rectangular ...

The ratio between the length and breadth of a rectangular field is 11:7. The cost of fencing it is Rs. 75,000. Find the dimensions of the field

Prove that x2 + y2 - 8x - 10y +39 = 0, If the points (5, 4) and (x, y) are ...

If the points (5, 4) and (x, y) are equidistant from the point (4, 5), prove that x 2 + y 2 - 8x - 10y +39 = 0. Ans :   AP = PB AP 2 = PB 2 (5 - 4) 2 + (4 - 5) 2 = (x

Probability, An unbiased die is tossed twice .Find the probability of getti...

An unbiased die is tossed twice .Find the probability of getting a 4,5,6 on the first toss and a 1,2,3,4 on the second toss

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