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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.
A publishing company is creating a book jacket for a newly published textbook. Determine the area of the book jacket, given that the front cover is 8 in wide by 11 in high, the bin
1. (a) Give an example of a function, f(x), that has an inflection point at (1, 4). (b) Give an example of a function, g(x), that has a local maximum at ( -3, 3) and a local min
how can you tell qhich trangle is sss,asa, sas, and aas s
1. The length of a rectangle is 2 times its width. The area of the rectangle is 72 square inches. Find the dimensions of the rectangle. 2. The length of a rec
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Financial institutions often create synthetic instruments out of existing instruments. In this case an investment bank plans to buy Treasury Bonds with 20-year maturities at their
Total Contribution per Year for next 10yeras =$1000+$800 =$1800 So Total Future fund Vaule =$1800*(1+1.073+power(1.073,2)+ power(1.073,2)+ power(1.073,3)+ power(1.073,4)+ power
Eddie mkes $15.75 per hour. Estimate how much Eddie will make per year if he works 40 hours per week and 50 weeks per year.
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What inequalities and intervals are? If it is given that a real number 'p' is not less than another real number 'q', we understand that either p should be equal to q or
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