  We present a coded caching framework using line graphs of bipartite graphs. A
clique cover of the line graph describes the uncached subfiles at users. A
clique cover of the complement of the square of the line graph gives a
transmission scheme that satisfies user demands. We then define a specific
class of such caching line graphs, for which the subpacketization, rate, and
uncached fraction of the coded caching problem can be captured via its graph
theoretic parameters. We present a construction of such caching line graphs
using projective geometry. The presented scheme has a rate bounded from above
by a constant with subpacketization level $q^{O((log_qK)^2)}$ and uncached
fraction $\Theta(\frac{1}{\sqrt{K}})$, where $K$ is the number of users and $q$
is a prime power. We also present a subpacketization-dependent lower bound on
the rate of coded caching schemes for a given broadcast setup.
