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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.
Question: Find Inverse Laplace Transform of the following (a) F(s) = (s-1)/(2s 2 +8s+13) (b) F(s)= e -4s /(s 2 +1) + (1/s 3 )
Given y = f(x) = x 2 + 2x +3 a) Use the definitional formula given below to find the derivative of the function. b) Find the value of the derivative at x = 3.
3x+y=9 5x-y=7
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