  Dynamical quantum phase transitions, encompassing phenomena like many-body
localization transitions and measurement-induced phase transitions, are often
characterized and identified through the analysis of quantum entanglement.
Here, we highlight that the dynamical phases defined by entanglement are
further enriched by complexity. We investigate both the entanglement and Krylov
complexity for fermions on regular graphs, which can be implemented by systems
like $^6$Li atoms confined by optical tweezers. Our investigations unveil that
while entanglement follows volume laws on both types of regular graphs with
degree $d = 2$ and $d = 3$, the Krylov complexity exhibits distinctive
behaviors. We analyze both free fermions and interacting fermions models. In
the absence of interaction, both numerical results and theoretical analysis
confirm that the dimension of the Krylov space scales as $D\sim N$ for regular
graphs of degree $d = 2$ with $N$ sites, and we have $D\sim N^2$ for $d = 3$.
The qualitative distinction also persists in interacting fermions on regular
graphs. For interacting fermions, our theoretical analyses find the dimension
scales as $D\sim 4^{N^\alpha}$ for regular graphs of $d = 2$ with
$0.38\leq\alpha\leq0.59$, whereas it scales as $D\sim 4^N$ for $d = 3$. The
distinction in the complexity of quantum dynamics for fermions on graphs with
different connectivity can be probed in experiments by measuring the
out-of-time-order correlators.
