Which mutation classes of quivers have constant number of arrows?

Abstract: We classify the connected quivers with the property that all the quivers in their mutation class have the same number of arrows. These are the ones having at most two vertices, or the ones arising from triangulations of marked bordered oriented surfaces of two kinds: either surfaces with non-empty boundary having exactly one marked point on each boundary component and no punctures, or surfaces without boundary having exactly one puncture. This combinatorial property has also a representation-theoretic counterpart: to each such quiver there is a naturally associated potential such that the Jacobian algebras of all the QP in its mutation class are derived equivalent.