On digraphs determined by their singular values

TL;DR

This paper computes singular values and trace norms of Laplacian matrices for specific digraphs and investigates which digraphs are uniquely determined by their singular values.

Contribution

It provides explicit formulas for singular values and trace norms of Laplacian matrices for certain digraphs and characterizes digraphs determined by their singular values.

Findings

Singular values and trace norms are computed for directed paths, cycles, and stars.

Directed paths, cycles, and stars are determined by their Laplacian and signless Laplacian singular values.

These digraphs are not uniquely determined by their adjacency singular values.

Abstract

Let $D$ be an digraph of order $n$ with adjacency matrix $A(D)$ and outdegree matrix $\Delta^+=\Delta^+(D)$. Then the Laplacian and signless Laplacian matrices of $D$ are respectively defined as $L(D)=\Delta^+-A(D)$ and $Q(D)=\Delta^++A(D)$. 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.