  There is a recent exciting line of work in distributed graph algorithms in
the $\mathsf{CONGEST}$ model that exploit expanders. All these algorithms so
far are based on two tools: expander decomposition and expander routing. An
$(\epsilon,\phi)$-expander decomposition removes $\epsilon$-fraction of the
edges so that the remaining connected components have conductance at least
$\phi$, i.e., they are $\phi$-expanders, and expander routing allows each
vertex $v$ in a $\phi$-expander to very quickly exchange $\text{deg}(v)$
messages with any other vertices, not just its local neighbors.
  In this paper, we give the first efficient deterministic distributed
algorithms for both tools. We show that an $(\epsilon,\phi)$-expander
decomposition can be deterministically computed in $\text{poly}(\epsilon^{-1})
n^{o(1)}$ rounds for $\phi = \text{poly}(\epsilon) n^{-o(1)}$, and that
expander routing can be performed deterministically in
$\text{poly}(\phi^{-1})n^{o(1)}$ rounds. Both results match previous bounds of
randomized algorithms by [Chang and Saranurak, PODC 2019] and [Ghaffari, Kuhn,
and Su, PODC 2017] up to subpolynomial factors.
  Consequently, we derandomize existing distributed algorithms that exploit
expanders. We show that a minimum spanning tree on $n^{o(1)}$-expanders can be
constructed deterministically in $n^{o(1)}$ rounds, and triangle detection and
enumeration on general graphs can be solved deterministically in $O(n^{0.58})$
and $n^{2/3 + o(1)}$ rounds, respectively.
  We also give the first polylogarithmic-round randomized algorithm for
constructing an $(\epsilon,\phi)$-expander decomposition in
$\text{poly}(\epsilon^{-1}, \log n)$ rounds for $\phi = 1 /
\text{poly}(\epsilon^{-1}, \log n)$. The previous algorithm by [Chang and
Saranurak, PODC 2019] needs $n^{\Omega(1)}$ rounds for any $\phi\ge
1/\text{poly}\log n$.
