Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On digraphs determined by their singular values

Mushtaq A. Bhat, Peer Abdul Manan

Abstract

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.

On digraphs determined by their singular values

Abstract

Let be an digraph of order with adjacency matrix and outdegree matrix . Then the Laplacian and signless Laplacian matrices of are respectively defined as and . In this paper, we compute singular values and an exact formula for the trace norm of Laplacian matrices of the directed path , the directed cycle and all orientations of a star. We show that for a bipartite digraph , the matrices and 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 , the directed cycle and oriented star are determined by their Laplacian and signless Laplacian singular values but are not determined by their adjacency singular values.

Paper Structure

This paper contains 3 sections, 7 theorems, 69 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Singular values and trace norm of Laplacian and signless Laplacian matrices
  3. Digraphs determined by Laplacian and signless Laplacian singular values

Key Result

Lemma 1.1

hj (Schur complement) Let $M \in {M}_{p+q}(\mathbb{R})$ be a block matrix If $D$ is invertible, then the Schur complement of $D$ in $M$ is the $p \times p$ matrix defined by Also, the determinant satisfies $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: An oriented Star $\vec{S}_n(x,y)$
  • Figure 2: Oriented trees on 4 vertices with outdegree sequence $[2, 1, 0, 0]$.
  • Figure 3: $U_1$, unicyclic digraph with outdegree sequence $[1^4]$
  • Figure 4: Possible subdigraphs $U_2, U_3, U_4$ and $T_4$ for $n \ge 5$.
  • Figure 5: Oriented trees with outdegree sequence $[1^4, 0]$.

Theorems & Definitions (7)

  • Lemma 1.1
  • Lemma 1.2
  • Theorem 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 3.1