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

Example of inverse matrix, Determine the inverse of the following matrix, i...

Determine the inverse of the following matrix, if it exists. We first form the new matrix through tacking onto the 3 x 3 identity matrix to this matrix.  It is, We

Implicit differentiation, Implicit Differentiation : To this instance w...

Implicit Differentiation : To this instance we've done quite a few derivatives, however they have all been derivatives of function of the form y = f ( x ) .  Unluckily not all

Tangent, construction of tangent when center not known

construction of tangent when center not known

Calculus, I need help with my calculus work

I need help with my calculus work

Build a fine automaton which accept all words, Build a Fine Automaton which...

Build a Fine Automaton which accept all words which have different first and last letters (that is if the word starts with an "a" to be accepted it should end with "b" and vice ver

Boundary value problem, solve the in-homogenous problem where A and b are c...

solve the in-homogenous problem where A and b are constants on 0 ut=uxx+A exp(-bx) u(x,0)=A/b^2(1-exp(-bx)) u(0,t)=0 u(1,t)=-A/b^2 exp(-b)

The sum of the clock, how many times In a 12 hour period will he numbers ad...

how many times In a 12 hour period will he numbers add up to 6? (hint 3:00 is one answer0

#Regular Expression, Find the Regular Grammar for the following Regular Exp...

Find the Regular Grammar for the following Regular Expression: a(a+b)*(ab*+ba*)b.

Weighted mean-progression, Weighted mean - It is the mean which employ...

Weighted mean - It is the mean which employs arbitrarily given weights - This is a useful measure especially whereas assessment is being done yet the situation prevailing a

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