Enumeration of the degree sequences of non-separable graphs and connected graphs

In 1962, S. L. Hakimi proved necessary and sufficient conditions for a given sequence of positive integers d 1 , d 2 , … , d n to be the degree sequence of a non-separable graph or that of a connected graph. Our goal in this note is to utilize these results to prove closed formulas for the functions d n s ( 2 m ) and d c ( 2 m ) , the number of degree sequences with degree sum 2 m representable by non-separable graphs and connected graphs (respectively). Indeed, we give both generating function proofs as well as bijective proofs of the following identities: d n s ( 2 m ) = p ( 2 m ) − p ( 2 m − 1 ) − ∑ j = 0 m − 2 p ( j ) and d c ( 2 m ) = p ( 2 m ) − p ( m − 1 ) − 2 ∑ j = 0 m − 2 p ( j ) where p ( j ) is the number of unrestricted integer partitions of j .