Year: 2026
Type: preprint
Abstract: For every $n$ and $r$ with $1\le r$ and $\binom{r+1}{2}\le n$, we construct a connected unweighted graph on $n$ vertices whose Max-Cut semidefinite relaxation has a unique optimal solution of rank $r$. The main examples are the Kneser graphs $K(m,2)$, whose unique optima have rank $m-1$. A six-vertex wheel supplies rank three, and attaching pendant vertices preserves both rank and uniqueness. For dense random graphs, we also prove a polynomial lower bound on the optimal rank.
Source: arXiv (Cornell University)
Authors: Bhardwaj, Avinash, Narayanan, Vishnu
Institution: Indian Institute of Technology Bombay
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223996004