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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.
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R={(r, ?):1=r= 2cos? ,-p/3= ? =p/3
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Determine the Projection of b = (2, 1, -1) onto a = (1, 0, -2) There is a requirement of a dot product and the magnitude of a. a → • b → = 4 ||a
A 90% acid solution is mixed with a 97% acid solution to obtain 21 litres of a 95% solution. Findout the quantity of every solutions to get the resultant mixture.
if 2 ballons cost 12 coins,use equivelent ratios to see how many coins 8 ballons would cost
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