Friday, February 12, 2016

Love and Hatred

I love you when you integrate me
But for your partitioning of me finer and finer.
I hate you for differentiating me
Though you wait for my limits.

Sum of the Squares of the Degrees

One of the most fundamental results in graphs is that the sum of the degrees of all the vertices of a graph is twice the number of edges in it.


This is is the famous Handshaking Theorem.
It is called Handshaking Theorem due to the analogy of the involvement of two hands in every handshake with that of the two vertices in every relation (edge).

Hence, the total number of relations of the persons (degrees of  the vertices) in a graph is double the number of relations (edges) in it.

What happens if we add the squares of the degrees of all the vertices in a simple graph?

It is found out that the sum of the squares of all vertices in a graph is the the sum of the degrees of the adjacent vertices of all the vertices the graph it self.


Why is it so? It is because, the square of the degree of a vertex is same as adding the degree of a vertex, the degree number of times.
                                                         n2=n+n+n+...+n (n times)
But one can easily verify that every vertex has exactly its degree number of adjacent vertices. Hence, when we sum the degrees of  the adjacent vertices of all the vertices of a graph, every degree is added that degree number of times. This proves the result.

We can also observe that if we add all the degrees and degree squares of all the vertices of a graph, then it is same as summing up all the degrees of all vertices in the closed neighbourhood of all the vertices of that graph.

Monday, February 08, 2016

The First Twenty Five Primes

Every prime number is a positive integer that is divisible by 1 and itself only. There are infinitely many prime numbers. Prime numbers are peculiar numbers which do not have a common predictable property other than that of the indivisibility. Because they are indivisible, the fundamental theorem of arithmetic is that they are the building blocks of numbers. That is to say that any positive integer greater than 1 can be written uniquely as the product of prime numbers.

There are 25 primes below 100. This is a method to remember all the first 25 primes.
We begin with 25 itself.
The first and third primes are 2 and 5 respectively. They are prime numbers but no other prime number would have 2 and 5 as its end digits
Now, 5+2 and 5-2 give the other two single digits primes. i.e., 7 and 3.
Hence, we have all the single digit primes as 2, 3, 5 and 7.

All primes with at least two digits would end only in 1, 3, 7 or 9.
We list all the two digit primes that end in 1.
Wen use 2 and 5 once again for that.
Splitting 2, we get the first two digit prime, 11.
Then, 1+2=3, 1+3=4, 1+5=6 and 1+6=7 are the first digits of the remaining primes.
Hence, the two digit primes are 11, 31, 41, 61 and 71. Thus we get all primes that end in 1.
A total of 9 primes, we have found out so far.

Now, we need to list all the primes that end in 3.
Adding the digits of the primes that end in 1, we get 1+1=2, 3+1=4, 4+1=5, 6+1=7 and 7+1=8.
Appending 3 to the right of the above numbers, we get the two digit primes that end in 3.
They are 23, 43, 53, 73 and 83. Hence, excluding 3, we got 13 primes. Of course, 13 is also a prime number.

What left are the primes that end in 7 or 9. But, this is very easy.
List all the digits in numbers from 1 to 10.
They are 1, 2, 3, 4, 5, 6, 7, 8, 9, 1 and 3.
Appending 7 or 9 as the case may be(by avoiding the trivial composite numbers we get,
17, 29, 37, 47, 59, 67, 79, 89, 97, 19 and 07.
These are the 11 prime numbers that end in 7 or 9.
Hence all the prime numbers below 100 are:
  1. 2 and 5
  2. 11, 31, 41, 61 and 71
  3. 3, 13, 23, 43, 53, 73 and 83.
  4. 7, 17, 37, 47, 67 and 97
  5. 19, 29, 59, 79 and 89.