  Due to the applications in network coding, subspace codes and designs have
received many attentions. Suppose that $k\mid n$ and $V(n,q)$ is an
$n$-dimensional space over the finite field $\mathbb{F}_{q}$. A $k$-spread is a
$\frac{q^n-1}{q^k-1}$-set of $k$-dimensional subspaces of $V(n,q)$ such that
each nonzero vector is covered exactly once. A partial $k$-parallelism in
$V(n,q)$ is a set of pairwise disjoint $k$-spreads. As the number of
$k$-dimensional subspaces in $V(n,q)$ is ${n \brack k}_{q}$, there are at most
${n-1 \brack k-1}_{q}$ spreads in a partial $k$-parallelism.
  By studying the independence numbers of Cayley graphs associated to a special
type of partial $k$-parallelisms in $V(n,q)$, we obtain new lower bounds for
partial $k$-parallelisms. In particular, we show that there exist at least
$\frac{q^{k}-1}{q^{n}-1}{n-1 \brack k-1}_q$ pairwise disjoint $k$-spreads in
$V(n,q)$.
