Littlewood-Richardson coefficients and the eigenvalues of integral line graphs

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 $2ω(G)$, where $ω(G)$ 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 $-2$.

Theorems & Definitions (8)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Remark 1.5
• Corollary 1.6
• Theorem 1.7
• Lemma 2.1