The Second Largest Eigenvalue of Stiffness Matrices of Normalized Complete Frameworks

Let $R(G,p)$ be the normalized rigidity matrix of a framework $(G,p)$ in $\mathbb R^d$, and let $$L(G,p)=R(G,p)R(G,p)^{T}$$ be the associated stiffness matrix. We study the extremal eigenvalues of $L(K_n,p)$ for complete frameworks whose vertices lie on the unit sphere and have centroid at the origin. Our main result shows that, whenever $d\ge2$ and the image of $p$ contains at least three distinct points, the second largest eigenvalue of $L(K_n,p)$ is exactly $n/2$. This settles the eigenvalue part of a conjecture of Lew et al. [Israel J. Math. 256, 2023]. We further construct an infinite family of examples, given by regular polygons embedded in a two-dimensional subspace, for which the eigenvalue $n/2$ has multiplicity $2n-4$. Consequently, the multiplicity predicted in the conjecture is not correct in general. Our results reveal a dichotomy: the value of the second largest eigenvalue is universal, while its multiplicity is sensitive to the geometry of the underlying point configuration.