Peer Abdul Manan | Searcharxiv

arXiv2026

On singular values and trace norm of signed digraphs

Let $S = (D, σ)$ be a signed digraph, where $D=(\mathcal{V},\mathcal{A})$ is the underlying digraph of $S$ and $σ:\mathcal{A}\rightarrow\{-1,+1\}$ is a sign function. In this paper, we study the adjacency singular values of signed digraphs. We provide a necessary and sufficient condition for the singular values of a digraph to remain invariant under signing. We study rank of signed digraphs and determine signed digraphs with rank one. As an application of this result, we obtain a lower bound for the trace norm of signed digraphs and characterize the extremal signed digraphs.

math.CO↗

arXiv2026

On digraphs determined by their singular values

Let $D$ be an digraph of order $n$ with adjacency matrix $A(D)$ and outdegree matrix $Δ^+=Δ^+(D)$. Then the Laplacian and signless Laplacian matrices of $D$ are respectively defined as $L(D)=Δ^+-A(D)$ and $Q(D)=Δ^++A(D)$. In this paper, we compute singular values and an exact formula for the trace norm of Laplacian matrices of the directed path $\overrightarrow{P_n}$, the directed cycle $\overrightarrow{C_n}$ and all orientations of a star. We show that for a bipartite digraph $D$, the matrices $L(D)$ and $Q(D)$ have same singular values and use this to compute the singular values and trace norm of signless Laplacian matrices. We study the problem of determination of digraphs by their singular values and prove the directed path $\overrightarrow{P_n}$, the directed cycle $\overrightarrow{C_n}$ and oriented star $\overrightarrow{S}_n(n-1,0)$ are determined by their Laplacian and signless Laplacian singular values but are not determined by their adjacency singular values.

math.CO↗

arXiv2024

Variation in $α$ trace norm of a digraph by deletion of a vertex or an arc and its applications

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.

math.CO↗

arXiv2024

Bounds for the trace norm of $A_α$ matrix of digraphs

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.

math.CO↗