There are two alternative definitions of discrete connections on triangulated manifolds. The most known one associates a group element to each edge. An alternative approach uses first-order operators on simplexes of higher dimension. We show that in dimension two such connections are associated with self-adjoint second order operators, and the self-adjointness is equivalent to existence of two factorizations. We also show that Laplace transformations can be interpreted as the star-triangle transformation used in electrical circuits.