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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.
ABC is a right triangle right-angled at C and AC=√3 BC. Prove that ∠ABC=60 o . Ans: Tan B = AC/BC Tan B = √3 BC/BC Tan B =√3 ⇒ Tan B = Tan 60 ⇒ B = 60
Example: Find out the radius of convergence for the following power series. Solution : Therefore, in this case we have, a n = ((-3) n )/(n7 n+1 ) a n+1 = (
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How many ways can 4 DVDs be arranged on a shelf? Solution: There are 4 ways to choose the first DVD, 3 ways to choose the second, 2 ways to choose the third and 1 way to choo
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f(x)=5x^-6 on the interval [1,infinity)
Unit Vector and Zero Vectors Unit Vector Any vector along with magnitude of 1, that is || u → || = 1, is called a unit vector. Zero Vectors The vector w → = (
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