  We study the lattice of extensions of four-valued Belnap--Dunn logic, called
super-Belnap logics by analogy with superintuitionistic logics. We describe the
global structure of this lattice by splitting it into several subintervals, and
prove some new completeness theorems for super-Belnap logics. The crucial
technical tool for this purpose will be the so-called antiaxiomatic (or
explosive) part operator. The antiaxiomatic (or explosive) extensions of
Belnap--Dunn logic turn out to be of particular interest owing to their
connection to graph theory: the lattice of finitary antiaxiomatic extensions of
Belnap--Dunn logic is iso\-morphic to the lattice of upsets in the homomorphism
order on finite graphs (with loops allowed). In particular, there is a
continuum of finitary super-Belnap logics. Moreover, a non-finitary
super-Belnap logic can be constructed with the help of this isomorphism. As
algebraic corollaries we obtain the existence of a continuum of antivarieties
of De Morgan algebras and the existence of a prevariety of De Morgan algebras
which is not a quasivariety.
