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

Find regular grammar for given regular expression, find regular grammar for...

find regular grammar for the following regular expression: a(a+b)*(ab* +ba*)b

Example of developing an understanding, I gave my niece a whole heap of bea...

I gave my niece a whole heap of beads and showed her how to divide it up into sets of 10 beads each. Then I showed her how she could lay out each set of I0 beads in a line, and cal

Example of decimal to fraction conversion, Example of Decimal to Fraction C...

Example of Decimal to Fraction Conversion: Example: Convert 18.82 to a mixed number. Solution: Step 1:            18.82 is 18 and 82 hundredths. 18.82 = 18(8

Differential Equations, 1.Verify Liouville''s formula for y "-y" - y'' + y ...

1.Verify Liouville''s formula for y "-y" - y'' + y = 0 in (0, 1) ? 2.Find the normalized differential equation which has {x, xex} as its fundamental set. 3.6Find the general soluti

Sketch the graph f ( x ) = - x5 + (5/2 )x4 + (40/3) x3 + 5, Sketch the grap...

Sketch the graph of the below function. f ( x ) = - x 5 + (5/2 )x 4 + (40/3) x 3 + 5 Solution : Whenever we sketch a graph it's good to have a few points on the graph to

Least common denominator of rational expression, Perform the denoted operat...

Perform the denoted operation.                    (4/6x 2 )-(1/3x 5 )+(5/2x 3 ) Solution For this problem there are coefficients on each of term in the denominator thus

Sequence-or-series, in and ap 1,2,3,4,5,6,7,8,9 11,12,13,14,15,16,17,18,19...

in and ap 1,2,3,4,5,6,7,8,9 11,12,13,14,15,16,17,18,19 and like that nonzzero digit find tn Solution) First break the ''n'' number in terms of 10''s power. For e.g if n=3259 wri

Quadratic equation whose roots are real, Write the quadratic equation whose...

Write the quadratic equation whose roots are real and non conjugate Ans)  x^2-x+6=0 ...roots are real and non conjugate

Find the sum-of-products expression for the function, Find the sum-of-produ...

Find the sum-of-products expression for subsequent function,  F (x,y,z) = y + Z‾ Ans: The sum of the product expression for the following function f is DNF (disjunc

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