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(ns)(2m) and d(c)(2m), the number of degree sequences with degree sum 2m representable by nonseparable graphs and connected graphs (respectively). Indeed, we give both generating function proofs as well as bijective proofs of the following identities: dns(2m) = p(2m) - p(2m - 1) - Sigma(m-2)(j=0)p(j) and d(c)(2m) = p(2m) - p(m - 1) - 2 Sigma(m-2)(j=0)p(j) where p(j) is the number of unrestricted integer partitions of j. (C) 2008 Elsevier Ltd. All rights reserved.