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Spectra of Corona Products of Digraphs

Michael Cavers, Farzad Maghsoudi, Babak Miraftab

TL;DR

The paper extends corona products to directed graphs by defining vertex- and arc-corona constructions and studies their adjacency, Laplacian, and signless Laplacian spectra via digraph coronals. It develops A-, L-, and Q-coronal tools and uses them, together with Schur complement and equitable-partition techniques, to derive compact formulas for the spectra of digraph coronas, including explicit arc-corona expressions. It connects coronals to complements, provides closed forms for families such as directed paths and cycles, and identifies special cases (out-regular, symmetric) where the spectra are completely determined by the base spectra. These results generalize classical graph corona theory to digraphs and enable efficient spectral analysis of complex digraph constructions.

Abstract

Two types of corona products for simple directed graphs are introduced, extending the classical notions from the undirected setting: the vertex-corona and the arc-corona. Their structural and spectral properties are investigated through the use of digraph coronals, with particular emphasis on the adjacency, Laplacian, and signless Laplacian spectra. Finally, the coronals corresponding to these three matrices are computed for several families of digraphs.

Spectra of Corona Products of Digraphs

TL;DR

The paper extends corona products to directed graphs by defining vertex- and arc-corona constructions and studies their adjacency, Laplacian, and signless Laplacian spectra via digraph coronals. It develops A-, L-, and Q-coronal tools and uses them, together with Schur complement and equitable-partition techniques, to derive compact formulas for the spectra of digraph coronas, including explicit arc-corona expressions. It connects coronals to complements, provides closed forms for families such as directed paths and cycles, and identifies special cases (out-regular, symmetric) where the spectra are completely determined by the base spectra. These results generalize classical graph corona theory to digraphs and enable efficient spectral analysis of complex digraph constructions.

Abstract

Two types of corona products for simple directed graphs are introduced, extending the classical notions from the undirected setting: the vertex-corona and the arc-corona. Their structural and spectral properties are investigated through the use of digraph coronals, with particular emphasis on the adjacency, Laplacian, and signless Laplacian spectra. Finally, the coronals corresponding to these three matrices are computed for several families of digraphs.

Paper Structure

This paper contains 11 sections, 27 theorems, 47 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coronals of matrices and their computations
  4. Complements of digraphs via coronals
  5. Computations of coronals
  6. Vertex Corona
  7. Arc Corona
  8. Adjacency spectrum of arc coronas
  9. Laplacian spectrum of arc coronas
  10. Signless Laplacian spectrum of arc coronas
  11. Further Research

Key Result

Lemma 1.1

BrualdiMR2312328[lemma]bb Let $D$ be a digraph and $\mathcal{L}(D)$ its line digraph. Then $A(D) = B_{\rm out}(D) B_{\rm in}(D)^T$ and $A(\mathcal{L}(D)) = B_{\rm in}(D)^T B_{\rm out}(D)$.

Figures (3)

  • Figure 1: The forward-vertex-corona, backward-vertex-corona and symmetric-vertex-corona of the directed cycle $C_3$ (indicated by gray arcs) and directed path $P_2$.
  • Figure 2: Possible connections between an arc $uv$ in $D_1=P_2$ and a vertex $w$ in $D_2=P_1$ in forming the arc corona. In (d), the symmetric edge $uv$ adds one copy of $D_2$ to the symmetric-arc-corona.
  • Figure 3: The forward-vertex-corona, backward-vertex-corona and symmetric-vertex-corona of the directed cycle $C_3$ (indicated with gray arcs) and directed path $P_2$.

Theorems & Definitions (51)

  • Lemma 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • proof
  • ...and 41 more