The pulsational properties of ultra-massive DB white dwarfs with carbon-oxygen cores coming from single-star evolution

[Context] Ultra-massive white dwarfs are relevant for many reasons: Their role as type Ia supernova progenitors, the occurrence of physical processes in the asymptotic giant branch phase, the existence of high-field magnetic white dwarfs, and the occurrence of double white dwarf mergers. Some hydrogen-rich ultra-massive white dwarfs are pulsating stars and, as such, they offer the possibility of studying their interiors through asteroseismology. On the other hand, pulsating helium-rich ultra-massive white dwarfs could be even more attractive objects for asteroseismology if they were found, as they should be hotter and less crystallized than pulsating hydrogen-rich white dwarfs, something that would pave the way for probing their deep interiors. [Aims] We explore the pulsational properties of ultra-massive helium-rich white dwarfs with carbon-oxygen and oxygen-neon cores resulting from single stellar evolution. Our goal is to provide a theoretical basis that could eventually help to discern the core composition of ultra-massive white dwarfs and the scenario of their formation through asteroseismology, anticipating the possible future detection of pulsations in helium-rich ultra-massive white dwarfs. [Methods] We focus on three scenarios for the formation of helium-rich ultra-massive white dwarfs. First, we consider stellar models coming from two recently proposed single-star evolution scenarios for the formation of ultra-massive white dwarfs with carbon-oxygen cores that involve the rotation of the degenerate core after core helium burning and reduced mass-loss rates in massive asymptotic giant branch stars. Finally, we contemplate ultra-massive oxygen-neon core white-dwarf models resulting from standard single-star evolution. We compute the adiabatic pulsation gravity-mode periods for models in a range of effective temperatures, embracing the instability strip of average-mass pulsating helium-rich white dwarfs, and we compare the characteristics of the mode-Trapping