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Nuclear reactions are key to our understanding of stellar evolution,particularly the $^{12}\rm{C}\left({\alpha},{\gamma}\right)^{16}\!\rm{O}$ rate,which is known to significantly influence the lower and upper ends of the blackhole (BH) mass distribution due to pair-instability supernovae (PISNe).However, these reaction rates have not been sufficiently determined. We use the$\texttt{MESA}$ stellar evolution code to explore the impact of uncertainty inthe $^{12}\rm{C}\left({\alpha},{\gamma}\right)^{16}\!\rm{O}$ rate on PISNexplosions, focusing on nucleosynthesis and explosion energy by considering thehigh resolution of the initial mass. Our findings show that the mass ofsynthesized radioactive nickel ($^{56}{\rm Ni}$) and the explosion energyincrease with $^{12}\rm{C}\left({\alpha},{\gamma}\right)^{16}\!\rm{O}$ rate forthe same initial mass, except in the high-mass edge region. With a high (abouttwice the $\texttt{STARLIB}$ standard value) rate, the maximum amount of nickelproduced falls below 70 $M_\odot$, while with a low rate (about half of thestandard value) it increases up to 83.9 $M_\odot$. These results highlight thatcarbon burning plays a crucial role in PISNe by determining when a starinitiates expansion. The initiation of expansion competes with collapse causedby helium photodisintegration, and the maximum mass that can lead to anexplosion depends on the$^{12}\rm{C}\left({\alpha},{\gamma}\right)^{16}\!\rm{O}$ reaction rate.

Impacts of the $^{12}\rm{C}\left({\alpha},{\gamma}\right)^{16}\!\rm{O}$  reaction rate on $^{56}{\rm Ni}$ nucleosynthesis in pair-instability  supernovae