TITLE: Degree sequence of connected graphs
QUESTION [8 upvotes]: Given graph degree sequence . What is the condition that it can be degree sequence of connected graph. Can anyone please share link to theorem?

REPLY [3 votes]: Recently, there has been some new research on this problem. Delen and Cangul defined a new invariant for graphs (and degree sequences) called omega invariant which helps to determine many properties of all the realizations of a given degree sequence including connectedness, realizability, cyclicness, number of loops, multiple edges, chords, maximum cycle length etc., see [Delen, S., Cangul, I. N., A New Graph Invariant, Turkish Journal of Analysis and Number Theory, 6 (1), (2018), 30-33 DOI: 10.12691/tjant-6-1-4] for the definition and fundamental properties and [212)   Delen, S., Cangul, I. N., Extremal Problems on Components and Loops in Graphs, Acta Mathematica Sinica, English Series, 35 (2) (2019), 161-171,  https://doi.org/10.1007/s10114-018-8086-6, WOS:000458139300011 (https://link.springer.com/content/pdf/10.1007%2Fs10114-018-8086-6.pdf)] for some properties. Some results are as follows: If omega of a degree sequence is less than or equal to -4, all realizations of it must be disconnected. If omega is at least -2, then the realizations could be connected or disconnected (that is, DS is potentially connected). If omega = -2, then every connected realization must be acyclic (tree).If omega is non-negative, all realizations are cyclic, etc... 
This new invariant also helps to prove many existing results like Edmonds' theorem.