Littlewood-Richardson coefficients and the eigenvalues of integral line graphs

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 2ω(G), where ω(G) is the clique number of G. Also using Horn's inequalities, we show that for every odd integer k≥ 19, a non-complete k-regular Ramanujan graph has an eigenvalue less than -2.