Degree sequence of connected graphs Given graph degree sequence . What is the condition that it can be degree sequence of connected graph. Can anyone please share link to theorem?

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. 
