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

Numerical integration - simpson rule, (1)Derive, algebraically, the 2nd ord...

(1)Derive, algebraically, the 2nd order (Simpson's Rule) integration formula using 3 equally spaced sample points, f 0 ,f 1 ,f 2 with an increment of h. (2) Using software such

Hello, I am here to tell you, Alex has a cold.

I am here to tell you, Alex has a cold.

Word problem in algebra, robin runs 5 kilometers around the campus in the s...

robin runs 5 kilometers around the campus in the same length of time as he can walk 3 kilometers from his house to school. If he runs 4 kilometers per hour faster than he walks, ho

NUMERABILITY, AFIGURE THIS OUT(3) (14) (17) (20) (25)= 8 WHAT ARE THE PROC...

AFIGURE THIS OUT(3) (14) (17) (20) (25)= 8 WHAT ARE THE PROCEDURES (-)(+)(x)(div) BETWEEN EACH NUMBER TO COME UP WITH 8 ?sk question #Minimum 100 words accepted#

Vectors, |a.x|=1 where x = i-2j+2k then calculate a

|a.x|=1 where x = i-2j+2k then calculate a

Determine the probability - conditional probability, A bag of 28 tulip bulb...

A bag of 28 tulip bulbs contains 12 red tulip bulbs, 9 yellow tulip bulbs, and 7 purple tulip bulbs. Two bulbs are selected without replacement. Determine, a) The probability th

Rotational symmetry .., write down the order of rotational symmetry of the ...

write down the order of rotational symmetry of the rectangle

Find the function and domain, Consider the function f(x) = x + 1/x 2 + 2x ...

Consider the function f(x) = x + 1/x 2 + 2x - 3. (a) Find f(2) and f(-2). (b) Find the domain of f(x). (c) Does the range include 1? Show your working. (d) Find and si

Evaluate this integral value, The base of a right cylinder is the circle in...

The base of a right cylinder is the circle in the xy -plane with centre O and radius 3 units. A wedge is obtained by cutting this cylinder with the plane through the y -axis in

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