How to write Δ and δ easily in word?

Press Alt and  += tab simultaneously.  Then a equation editor will appear like below

In the editor , type for \Delta for Δ and \delta for δ.

visulaization of network

Havel-Hakimi Algorithm: Find the existence of a simple graph from a given degree sequence

We can find out whether a simple graph exist corresponding to a given degree sequence using Havel-Hakimi Algorithm


Step 1: Put degree sequence in non-ascending order.
Step 2:  Remove highest degree (K)   
Step 3: Subtract 1 from next K entries
Step 4: Repeat step 1, 2 and 3 until

(i)                  If we get all zero entries ----> Simple graph exist

(ii)                If we get at least one negative entry --->   Simple graph does not exist

(iii)               Not enough degree or entries ---> Simple graph does not exist.

For example

(     1)    3,2,1,1,0  

       Solution: No graph exist
                     Reason: The sum of degrees of a graph is even

(     2)    3,2,1,0,0

      Solution:
          Step 1: It is already in non-ascending sequence.
          Step 2: We get the value of K=3.  After removal of highest degree, the sequence becomes (2,1,0,0)
          Step 3: We subtract 1 from K=3 entries (2,1,0) from the whole sequence (2,1,0,0). Then the sequence becomes (1,0,-1,0)
          Step 4: We stop here as we get one negative entry in the sequence.
                       Conclusion: no simple graph exists.

3)  8,7,7, 6, 4,2,1,1
       Solution: 
                

          Step 1: It is already in non-ascending sequence.

          Step 2: We get the value of K=8.  After removal of highest degree, the sequence becomes (7,7, 6, 4,2,1,1)

         Step 3: We have to subtract 1 from K=8 entries. But we have only 7 entries.

                        Conclusion: no simple graph exists.

4) 7,6,5,4,4,3,2,1

Solution: Simple graph exists. You will get all 0 entries if you follow the above procedure.