  We multiply two $n \times n$ matrices $S,T$ over semirings in the Congested
Clique model, where $n$ fully connected nodes communicate synchronously using
$O(\log n)$-bit messages, within $O(nz(S)^{1/3} nz(T)^{1/3}/n + 1)$ rounds of
communication, where $nz(A)$ denotes the number of non-zero elements in a
matrix $A$. By leveraging the sparsity of the input matrices, our algorithm
greatly reduces communication compared with general algorithms [Censor-Hillel
et al., PODC 2015], improving upon the state-of-the-art for matrices with
$o(n^2)$ non-zero elements. Our algorithm exhibits the additional strength of
surpassing previous solutions also when only one matrix is sparse. This allows
efficiently raising a sparse matrix to a power greater than 2. As applications,
we speed up 4-cycle counting and APSP in sparse graphs.
  Our algorithmic contribution is a new \emph{deterministic} method of
restructuring the input matrices in a sparsity-aware manner, which assigns each
node with element-wise multiplication tasks that are not necessarily
consecutive but are balanced, yielding communication-efficient multiplication.
  Moreover, this new deterministic method for restructuring matrices may be
used to restructure the adjacency matrix of input graphs, enabling faster
solutions for graph related problems. As an example, we present a new
deterministic algorithm which solves the triangle listing problem in
$O(m/n^{5/3} + 1)$ rounds, a complexity that was previously obtained by a
\emph{randomized} algorithm [Pandurangan et al., SPAA 2018] and matches the
lower bound of $\tilde{\Omega}(n^{1/3})$ when $m=n^2$ of [Izumi and Le Gall,
PODC 2017, Pandurangan et al., SPAA 2018].
  Our triangle listing algorithm implies triangle counting with the same
complexity of $O(m/n^{5/3} + 1)$ rounds, which is a \emph{cubic} improvement
over the previous $O(m^2/n^3)$-round algorithm [Dolev et al., DISC 2012].
