The McKay correspondence can be extended to multiply laced Dynkin diagrams, by using a "pair" of binary polyhedral groups. This is known as the Slodowy correspondence, named after Peter Slodowy – see . The ADE graphs and the extended (affine) ADE graphs can also be characterized in terms of labellings with certain properties, which can be stated in terms of the discrete Laplace operators or Cartan matrices. Proofs in terms of Cartan matrices may be found in . The affine ADE graphs are the only graphs that admit a positive labeling (labeling of the nodes by positive real numbers) with the following property: That is, they are the only positive functions with eigenvalue 1 for the discrete Laplacian (sum of adjacent vertices minus value of vertex) – the positive solutions to the homogeneous equation: Equivalently, the positive functions in the kernel of formula_30 The resulting numbering is unique up to scale, and if normalized such that the smallest number is 1, consists of small integers – 1 through 6, depending on the graph. The ordinary ADE graphs are the only graphs that admit a positive labeling with the following property: In terms of the Laplacian, the positive solutions to the inhomogeneous equation: The resulting numbering is unique (scale is specified by the "2") and consists of integers; for E8 they range from 58 to 270, and have been observed as early as . The elementary catastrophes are also classified by the ADE classification. The ADE diagrams are exactly the quivers of finite type, via Gabriel's theorem. There is also a link with generalized quadrangles, as the three non-degenerate GQs with three points on each line correspond to the three exceptional root systems "E"6, "E"7 and "E"8. The classes "A" and "D" correspond degenerate cases where the line set is empty or we have all lines passing through a fixed point, respectively. There are deep connections between these objects, hinted at by the classification; some of these connections can be understood via string theory and quantum mechanics. It was suggested that symmetries of small droplet clusters may be subject to an ADE classification. The minimal models of two-dimensional conformal field theory have an ADE classification. Arnold has subsequently proposed many further connections in this vein, under the rubric of "mathematical trinities", and McKay has extended his correspondence along parallel and sometimes overlapping lines.