Ultramassive Black Holes in the Most Massive Galaxies: M _BH–σ versus M _BH–R _b

[Abridged] We investigate the nature of relations between black hole (BH) mass ($M_{\rm BH}$) and central velocity dispersion ($\sigma$) and, for core-S\'ersic galaxies, size depleted core ($R_{\rm b}$). Our sample 144 galaxies with dynamically determined $M_{\rm BH}$ encompasses 24 thought to be products gas-poor mergers, reliably identified based on high-resolution HST imaging. For -- i.e., combining normal-core b} < 0.5 $ kpc) large-core \gtrsim 0.5$ kpc), we find that correlates remarkably well $R_{\rm b}$ such BH} \propto R_{\rm b}^{1.20 \pm 0.14}$ (rms scatter in log $\Delta_{\rm rms} \sim 0.29$ dex), confirming previous works same except three new ones. Separating into S\'ersic, S\'ersic jointly define a single log-linear BH}-\sigma$ relation \sigma^{ 4.88 0.29}$ 0.47$ dex, however, at high-mass end (four measured are offset upward from this by ($2.5-4) \times \sigma_{\rm s}$, explaining previously reported steepening massive galaxies. Large-core spheroids have magnitudes $M_{V} \le -23.50$ mag, half-light radii Re $>$ 10 kpc extremely $M_{*} \ge 10^{12}M_{\odot}$. Furthermore, these tend host ultramassive BHs 10^{10}M_{\odot}$) tightly connected their rather than $\sigma$. The less popular BH}-R_{\rm exhibits $\sim$ 62% BH}- \sigma$ relations.