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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.
Based upon the primary sources, describe at least three characteristics that mark the early modern world as distinctly different than the Medieval world that preceded it. You might
Find out the product of 5.2 × 10 3 and 6.5 × 10 7 . Write your answer in scientific notation. To multiply numbers written within scienti?c notation, multiply the ?rst numbers
01010011 01100101 01101101 01110000 01100101 01110010 00100000 01000110 01101001 00100001
1+2+3+78+980
There are only Chinese and Malay pupils in a hall.The ratio of the number of boys to the number of girls is 2:3.The ratio of the number of Chinese boys to the number of Malay boys
what is the lcm of 4, 6 ,18?
#question.x2-y2-4x-2y+3.
what is the expiremental probability that the next toss and spin will result in 3 and tail if 1-heads,53.2-heads,49.3-heads,54.1-tail,65.2-tails,71.3-tails,62
I have a log that is 1/3 in mud and the rest of it is 6 meters long. How long is the entire log?
Illustration 1 In a described exam the scores for 10 students were given as: Student Mark (x) |x-x¯| A 60
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