  This note describes necessary and sufficient conditions for a sequence of
positive integers to be the degree sequence of a connected simple graph.
Conditions are also given under which a sequence is necessarily connected i.e.
the sequence can only be realised as a connected graph. A matrix is introduced
whose non-empty entries partition the set of connected graphs. The note
concludes with a result relating the number of edges in a simple graph to the
connectedness of the graph.
