Closed-form degree formula for Γ(R, k) with k > 1

Determine a closed-form expression for the vertex degree(s) in the graphs Γ(R, k) for k > 1 when R is an irreducible simply-laced root system. Here Γ(R, k) has vertices given by sums of k-element strongly orthogonal subsets of roots in R, with edges joining two vertices when their difference is also such a sum. Extend the k = 1 case, where Γ(R, 1) is regular of degree 2(h − 2) in terms of the Coxeter number h, to obtain formulas for k > 1 (accounting for potential multiple Weyl orbits in non-regular cases).

Background

The paper defines Γ(R, k) with vertex set consisting of sums of k-element strongly orthogonal subsets (SOS) of a root system R, and edges when the difference of two vertices is also a vertex. For irreducible simply-laced R, Γ(R, 1) is shown to be regular of degree 2(h − 2), where h is the Coxeter number.

For k > 1, computations indicate that Γ(R, k) is regular for all simply-laced exceptional systems except (E7, 3), (E7, 4), and (E8, 4), which split into two Weyl orbits with distinct degrees. Despite this empirical structure, a general closed-form degree formula for k > 1 is not provided.