Known algorithms to go from a DFA to a regular expression
Two more constructions: Brzozowski-McCluskey aka state elimination [1], and Gaussian elimination in a system of equations using Arden's Lemma.  The best source on these is probably Jacques Sakarovitch's book [2].

[1] J. Brzozowski, E. McCluskey Jr., Signal ﬂow graph techniques for sequential circuit state diagrams, IEEE Transactions on Electronic 
Computers EC-12 (1963) 67–76.

[2] J. Sakarovitch, Elements of Automata Theory.  Cambridge University Press, 2009.