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

Proof integral function, Proof of: if f(x) > g(x) for a x b th...

Proof of: if f(x) > g(x) for a x b then a ∫ b  f(x) dx > g(x). Because we get f(x) ≥ g(x) then we knows that f(x) - g(x) ≥ 0 on a ≤ x ≤ b and therefore by Prop

I am mathematics expert, i want some assignment for earning i am mathemati...

i want some assignment for earning i am mathematics expert plz provide us mathematics assignment as soon as possible

Horizontal asymptotes, Horizontal asymptotes : Such as we can have vert...

Horizontal asymptotes : Such as we can have vertical asymptotes defined in terms of limits we can also have horizontal asymptotes explained in terms of limits. Definition

Find the integral of a function, We want to find the integral of a function...

We want to find the integral of a function at an arbitrary location x from the origin. Thus, where I(x=0) is the value of the integral for all times less than 0. (Essenti

Square and square root., the value of square root of 200multiplied by squar...

the value of square root of 200multiplied by square root of 5+

BIOMATH, Ask quHarvesting prevents the population size of a species from at...

Ask quHarvesting prevents the population size of a species from attaining its natural carrying capacity. We can add harvesting to the logistic model by assuming that the per capita

Solid Mensuration, The two sides of a triangle are 17 cm and 28 cm long, an...

The two sides of a triangle are 17 cm and 28 cm long, and the length of the median drawn to the third side is equal to 19.5 cm. Find the distance from an endpoint of this median to

Assemble the coefficient matrix and solve the linear system, Solve discrete...

Solve discrete harmonic mapping of a given surface patch (suppose the surface is genus-0 and with one boundary) 1. Map the boundary loop onto a unit rectangle using chord-length

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