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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.
Problem 1 Let ~x0 = A~x and y 0 = B~y be two 2 2 linear systems of ODE. (1) Suppose that A and B have the same purely imaginary eigenvalues. Prove that these systems are topologi
SUMMATION NOTATION Under this section we require to do a brief review of summation notation or sigma notation. We will start out with two integers, n and m, along with n a
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
#question.x2-y2-4x-2y+3.
Find all the local maximum and minimum values and saddle points of the function f(x, y) = x 2 - xy + y 2 + 9x - 6y + 10
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How do I solve step by step 7
Describe, in your own words, the following terms and give an example of each. Your examples are not to be those given in the lecture notes, or provided in the textbook. By the en
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Find out the next number in the subsequent pattern. 320, 160, 80, 40, . . . Each number is divided by 2 to find out the next number; 40 ÷ 2 = 20. Twenty is the next number.
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