In the paper "Sederberg, T. W., Zheng, J., Bakenov, A., & Nasri, A. (2003). T-splines and T-NURCCs. ACM transactions on graphics (TOG), 22(3), 477-484." eigenvalue is used to prove the continuity of geometry. It's my first time to find someone gives a relationship between those. I was wondering why we can get continuity of geometry based on eigenvalue. The following picture is the detail.
The smoothness (order of continuity) of a subdivision surface is related to the eigenvalues of the subdivision matrix. This is a well-known approach to smoothness analysis. If you search for "subdivision surface" and "eigenvalue", you'll find lots of references on the subject.
A good place to start might be this book by Peters and Rief.