Discrete Laplace Operator - ADE Classification

ADE Classification

Certain equations involving the discrete Laplacian only have solutions on the simply-laced Dynkin diagrams (all edges multiplicity 1), and are an example of the ADE classification. Specifically, the only positive solutions to the homogeneous equation:

in words,

"Twice any label is the sum of the labels on adjacent vertices,"

are on the extended (affine) ADE Dynkin diagrams, of which there are 2 infinite families (A and D) and 3 exceptions (E). The resulting numbering is unique up to scale, and if the smallest value is set at 1, the other numbers are integers, ranging up to 6.

The ordinary ADE graphs are the only graphs that admit a positive labeling with the following property:

Twice any label minus two is the sum of the labels on adjacent vertices.

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 (Bourbaki 1968).