Variation in $α$ trace norm of a digraph by deletion of a vertex or an arc and its applications
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915015987036160 |
|---|---|
| author | Bhat, Mushtaq A. Manan, Peer Abdul |
| author_facet | Bhat, Mushtaq A. Manan, Peer Abdul |
| contents | Let $D$ be a digraph of order $n$ with adjacency matrix $A(D)$. For $α\in[0,1)$, the $A_α$ matrix of $D$ is defined as $A_α(D)=αΔ^{+}(D)+(1-α)A(D)$, where $Δ^{+}(D)=\mbox{diag}~(d_1^{+},d_2^{+},\dots,d_n^{+})$ is the diagonal matrix of vertex outdegrees of $D$. Let $σ_{1α}(D),σ_{2α}(D),\dots,σ_{nα}(D)$ be the singular values of $A_α(D)$. Then the trace norm of $A_α(D)$, which we call $α$ trace norm of $D$, is defined as $\|A_α(D)\|_*=\sum_{i=1}^{n}σ_{iα}(D)$. In this paper, we study the variation in $α$ trace norm of a digraph when a vertex or an arc is deleted. As an application of these results, we characterize oriented trees and unicyclic digraphs with maximum $α$ trace norm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_07935 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Variation in $α$ trace norm of a digraph by deletion of a vertex or an arc and its applications Bhat, Mushtaq A. Manan, Peer Abdul Combinatorics 05C20, 05C50 Let $D$ be a digraph of order $n$ with adjacency matrix $A(D)$. For $α\in[0,1)$, the $A_α$ matrix of $D$ is defined as $A_α(D)=αΔ^{+}(D)+(1-α)A(D)$, where $Δ^{+}(D)=\mbox{diag}~(d_1^{+},d_2^{+},\dots,d_n^{+})$ is the diagonal matrix of vertex outdegrees of $D$. Let $σ_{1α}(D),σ_{2α}(D),\dots,σ_{nα}(D)$ be the singular values of $A_α(D)$. Then the trace norm of $A_α(D)$, which we call $α$ trace norm of $D$, is defined as $\|A_α(D)\|_*=\sum_{i=1}^{n}σ_{iα}(D)$. In this paper, we study the variation in $α$ trace norm of a digraph when a vertex or an arc is deleted. As an application of these results, we characterize oriented trees and unicyclic digraphs with maximum $α$ trace norm. |
| title | Variation in $α$ trace norm of a digraph by deletion of a vertex or an arc and its applications |
| topic | Combinatorics 05C20, 05C50 |
| url | https://arxiv.org/abs/2411.07935 |