Definition 1 (realization): Let $k$ be a positive integer and $G = (V, E, d)$ be a simple graph with $n$ vertices, undirected, connected, and with edge weights. Find a function $x: V \rightarrow \mathbb{R}^{k}$ such that:
$$\forall \{v_{i}, v_{j}\} \in E, \Vert x(v_{i}) - x(v_{j}) \Vert_{2} = d(v_{i}, v_{j}).$$
Definition 2 (Discretizable Molecular Distance Geometry Problem): Given $k = 3$, a simple undirected weighted graph $G = (V, E, d)$ where $d : E → \mathbb{R}^{+}$, an order on $V$ denoted by $v_{1}, \cdots, v_{n}$ such that:
- There is a valid realization for $v_{1}, v_{2}, v_{3}$;
- For every $v_{i}, i = 4, \cdots, n$ there exist at least $3$ immediately preceding vertices $v_{i-1}, v_{i-2}, v_{i-3}$, where $\{v_{i-3}, v_{i-2}, v_{i-1}\}$ is a 3-clique and
$$d(v_{i-3}, v_{i-2}) + d(v_{i-2}, v_{i-1}) > d(v_{i-3}, v_{i-1}).$$
Find a function $x: V \rightarrow \mathbb{R}^{3}$ such that
$$\forall {v_{i}, v_{j}} \in E, \Vert x(v_{i}) - x(v_{j}) \Vert = d_{v_{i}, v_{j}}.$$
My Question: There is a result that states that the number of solutions to the DMDGP is $2^{\vert S \vert}$, where
$$S = \{v_{l}; \not \exists \{v_{i}, v_{j}\} \in E \;\text{such that}\; i+3 < l \leq j\}$$
and it's called the set of symmetry vertices. This result is proved using Partial Reflection Operator Theory and Group Theory. My question is, is there a way to establish a connection between this result and the Spectral Graph Theory?