  We first describe a system of inequalities (Horn's inequalities) that
characterize eigenvalues of sums of Hermitian matrices.
  When we apply this system for integral Hermitian matrices, one can directly
test it by using Littlewood-Richardson coefficients.
  In this paper, we apply Horn's inequalities to analysis the eigenvalues of an
integral line graph $G$ of a connected bipartite graph. Then we show that the
diameter of $G$ is at most $2\omega(G)$, where $\omega(G)$ is the clique number
of $G$. Also using Horn's inequalities, we show that for every odd integer
$k\geq 19$, a non-complete $k$-regular Ramanujan graph has an eigenvalue less
than $-2$.
