Bounds for the trace norm of $A_α$ matrix of digraphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914936201936896 |
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| 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 find the singular values of some basic digraphs and characterize the digraphs $D$ with $\mbox{Rank}~(A_α(D))=1$. As an application of these results, we obtain a lower bound for the trace norm of $A_α$ matrix of digraphs and determine the extremal digraphs. In particular, we determine the oriented trees for which the trace norm of $A_α$ matrix attains minimum. We obtain a lower bound for the $α$ spectral norm $σ_{1α}(D)$ of digraphs and characterize the extremal digraphs. As an application of this result, we obtain an upper bound for the $α$ trace norm of digraphs and characterize the extremal digraphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02602 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds for the trace norm of $A_α$ matrix of digraphs 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 find the singular values of some basic digraphs and characterize the digraphs $D$ with $\mbox{Rank}~(A_α(D))=1$. As an application of these results, we obtain a lower bound for the trace norm of $A_α$ matrix of digraphs and determine the extremal digraphs. In particular, we determine the oriented trees for which the trace norm of $A_α$ matrix attains minimum. We obtain a lower bound for the $α$ spectral norm $σ_{1α}(D)$ of digraphs and characterize the extremal digraphs. As an application of this result, we obtain an upper bound for the $α$ trace norm of digraphs and characterize the extremal digraphs. |
| title | Bounds for the trace norm of $A_α$ matrix of digraphs |
| topic | Combinatorics 05C20, 05C50 |
| url | https://arxiv.org/abs/2409.02602 |