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arXiv · 2303.01304

Littlewood-Richardson coefficients and the eigenvalues of integral line graphs

Mahdi Ebrahimi
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Abstract

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 $-2$.

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Mahdi Ebrahimi. 2023-02-27. Littlewood-Richardson coefficients and the eigenvalues of integral line graphs. https://arxiv.org/abs/2303.01304

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