Lew–Noy–Plaza–Eberhardt conjecture on the second largest eigenvalue of normalized complete frameworks

Let $d\geq 3$, and let $p:[n]\to\mathbb{R}^d$ satisfy

$$\|p(i)\|=1$$

for all $i\in[n]$ and

$$\sum_{i=1}^n p(i)=0.$$

Assume that the image of $p$ has size at least $3$. Lew–Noy–Plaza–Eberhardt's conjecture. The second largest eigenvalue of $L(K_n,p)$ is

$$\frac{n}{2},$$

and its multiplicity is exactly $n-1$. This conjecture asks whether the lower bound from the preceding result is sharp and whether $n/2$ has an extremal spectral interpretation for stiffness matrices of complete frameworks under the stated spherical normalization; its resolution is not indicated here.