..ooo Sprint use the theorem that r E3 sun id degrees to the number of edges to

Question

..ooo Sprint use the theorem that r E3 sun id degrees to the number of edges to determine the number of edges in the graph A graph with5 vertices one degreee 4 three of degree 3 and one of the degree 1 Explain the similarities and diffeeences between Hamilton circuits and ruler circuits Determine how many Hamilton circuits a complete graph with 29 vertices has Sarah Katerniva is a high school student in Chicago. She will be going to college next year and is planning to visit the following campsuses. University of Wisconsin at Madison Harvard and Ohio state university. How many different ways can she visit each of these schools and return to her starting point in Chicago? A circuit cannot be both a Hamilton and an euler circuit true or false and why Two primes are prime. Inverse whose differences is a multiple of 4 true or false Not all perfect numbers end in 6 or 28 true or false Is the number 18 abundant or deficient Write the numbers as the sum of two primes number Determine weather or not one or more pairs of twin primes exist between the pair of numbers given. If so identify the twin primes 33 and 39 Two natural relativity prime numbers have at mostone common factor true or false Find the greatest common factor of the numbers in the group 30 and 36 Find the least common multiple of the numbers in the If a 13 inch wide rectangle is to approach the golden ratio what should its length be What is the 20th term of the Lucas sequence The following quotient where fn represents the nth term of the Fibonacci seguence approaches the

Explanation / Answer

Question1

According to sum ofdegrees of vertices theorem, for any graph , the sum of the degrees of the vertices is equal to twice the number of edges.

That is if d1,d2,d3,d4 are the degrees of 4 vertices of a graph, then twice the number of edges

2e = (d1+d2+d3+d4)

Therefore the number of edges, e = (d1+d2+d3+d4)/2

In the given graph, there is 1 vertex of degree 4, 1 vertex of degree 1 and 3 vertices of degree 3.

Hence the number of edges e = (4+1+ 3+3+3) / 2 = 14/ 2 = 7

Question 2

Similarities and differences between Hamiltonian and Euler circuits

In a Hamilton circuit each vertex is visited exactly only once, whereas in a Euler circuit, every edge is visited only once.

Hamilton circuit is NP-complete, where as Euler circuit is linear and the algorithm is very straightforward.

Question 3

Consider a complete graph with n vertices.

starting at vertex 1 we have (n – 1) choices of moving to vertex 2;

From vertex 2 we will have (n– 2) choices of moving to vertex 3;

From vertex 3 we have (n – 3) choices of moving to vertex 4;

continuing in this manner, returning to vertex 1 and multiplying these choices together we have:

(n – 1) x (n -2) x (n – 3) x … x 3 x 2 x 1 = (n – 1)!

(n – 1)! gives the total number of Hamilton circuits in a complete graph.

So in the given problem, number of vertices = 29.

Hence the total number of hamiltonian circuits = ( 29-1)! = 28!

Question 4

There are 2 ways by which she can visit these universities and return to chicago

1. Visit University of Wisconsin first and then visit Ohio state university .

2. Visit Ohio state university and then visit University of Wisconsin.

This is a permutaion problem to decxide the permutation of 2 items taking all at a time.

Here n Pr = 2 P 2 = 2!/(2-2)! = 2

Question 6

Question is not clear

Question 7

A Perfect number is a positive integer which is equal to the sum of its proper divisors.

The smallest perfect number is 6, which is the sum of 1, 2, and 3.

Other perfect numbers are 28, 496, and 8,128.

All perfect numbers end in 6 or 28

Hence the given statement is false

Question 5

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