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We first describe a system of inequalities (Horn's inequalities) thatcharacterize eigenvalues of sums of Hermitian matrices. When we apply thissystem for integral Hermitian matrices, one can directly test it by usingLittlewood-Richardson coefficients. In this paper, we employ this approach toanalysis the eigenvalues of an integral line graph $G$ with a connectedbipartite root graph. Then we show that the diameter of $G$ is at most$2\omega(G)$, where $\omega(G)$ is the clique number of $G$.

Littlewood-Richardson coefficients and the eigenvalues of integral line graphs