Liu–Li's minimum spectral radius conjecture for odd-order graphs

Let $n\geq 3$ be odd, and let $G$ be a graph in $\mathbb{G}_{n,\lfloor n/2\rfloor}$. Write $\rho(G)$ for its spectral radius and $T^{(n+3)/2}_{(n-3)/2,0}$ for the specified graph in this family. Liu–Li's conjecture. For every $G\in\mathbb{G}_{n,\lfloor n/2\rfloor}$,

$$\rho(G)\geq \rho\left(T^{\frac{n+3}{2}}_{\frac{n-3}{2},0}\right),$$

and equality holds if and only if

$$G\cong T^{\frac{n+3}{2}}_{\frac{n-3}{2},0}.$$

The conjecture identifies the unique graph minimizing the spectral radius among graphs of odd order $n$ in the indicated family; the source paper is titled as a disproof, so the assertion is refuted.