Littlewood-Richardson coefficients and the eigenvalues of integral line graphs
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929268087324672 |
|---|---|
| author | Ebrahimi, Mahdi |
| author_facet | Ebrahimi, Mahdi |
| contents | 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 $-2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_01304 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Littlewood-Richardson coefficients and the eigenvalues of integral line graphs Ebrahimi, Mahdi Combinatorics 05E10, 05C50, 05C76 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 $-2$. |
| title | Littlewood-Richardson coefficients and the eigenvalues of integral line graphs |
| topic | Combinatorics 05E10, 05C50, 05C76 |
| url | https://arxiv.org/abs/2303.01304 |