In 1962, Hartshorne proved that the dual graphs of an arithmetically Cohen-Macaulay scheme is connected. After establishing a correspondence between the languages of algebraic geometry, commutative algebra and combinatorics, we are going to refine Hartshorne's result and measure the connectedness of the dual graphs of certain projective schemes in terms of an algebro-geometric invariant of the projective schemes themselves, namely their Castelnuovo-Mumford regularity. Time permitting, we are also going to address briefly the inverse problem of Hartshorne's result, by showing that any connected graph is the dual graph of a projective curve with nice geometric properties. This is joint work with Bruno Benedetti and Matteo Varbaro.