  The present paper offers, in its first part, a unified approach for the
derivation of families of inequalities for set functions which satisfy
sub/supermodularity properties. It applies this approach for the derivation of
information inequalities with Shannon information measures. Connections of the
considered approach to a generalized version of Shearer's lemma, and other
related results in the literature are considered. Some of the derived
information inequalities are new, and also known results (such as a generalized
version of Han's inequality) are reproduced in a simple and unified way. In its
second part, this paper applies the generalized Han's inequality to analyze a
problem in extremal graph theory. This problem is motivated and analyzed from
the perspective of information theory, and the analysis leads to generalized
and refined bounds. The two parts of this paper are meant to be independently
accessible to the reader.
