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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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At time t an investor shorts a $1 face value zero coupon bond that matures at time T = t and uses the entire proceeds to purchase a zero coupon bond that matures at time
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We will be looking at solutions to the differential equation, in this section ay′′ + by′ + cy = 0 Wherein roots of the characteristic equation, ar 2 + br + c = 0 Those
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