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

Number of permutations of ''n'' dissimilar things , Finding the numbe...

Finding the number of Permutations of 'n' dissimilar things taken 'r' at a time:  After looking at the definition of permutations, we look at how to evolve a

Determine the other two sides of the triangle, The radius of the in circle ...

The radius of the in circle of a triangle is 4cm and the segments into which one side is divided by the point of contact are 6cm and 8cm.  Determine the other two sides of the tria

Graph, Now we need to discuss graphing an equation. The first question whic...

Now we need to discuss graphing an equation. The first question which we have to ask is what accurately is a graph of an equation?  A graph is the set of all the ordered pairs whos

Write down the first few terms of the sequences, Write down the first few t...

Write down the first few terms of each of the subsequent sequences. 1. {n+1 / n 2 } ∞ n=1 2. {(-1)n+1 / 2n} ∞ n=0 3. {bn} ∞ n=1, where bn = nth digit of ? So

Mechanical vibrations, This time we are going to take a look at an applicat...

This time we are going to take a look at an application of second order differential equations. It's now time take a look at mechanical vibrations. In exactly we are going to look

Question, What is a marketing plan

What is a marketing plan

Numercial analysis and computer techniques, write FORTRAN programme to gene...

write FORTRAN programme to generate prime numbers between 1 and 100

Mean value theorem function, Mean Value Theorem : Suppose f (x) is a funct...

Mean Value Theorem : Suppose f (x) is a function which satisfies both of the following. 1. f ( x )is continuous on the closed interval [a,b]. 2. f ( x ) is differentiable on

How mathematical ideas grow, HOW MATHEMATICAL IDEAS GROW :  In this sectio...

HOW MATHEMATICAL IDEAS GROW :  In this section we shall consider three aspects of the nature of mathematical ideas, namely, that they progress from concrete to abstract, from part

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