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On Spectrum of Neighbourhood Corona Product of Signed Graphs

Bishal Sonar, Satyam Guragain, Ravi Srivastava

TL;DR

The paper advances the spectral theory of signed graphs by introducing and analyzing the neighbourhood corona product $\Gamma_1*\Gamma_2$. It provides general closed-form relations for the adjacency, Laplacian, and signless Laplacian spectra of $\Gamma_1*\Gamma_2$ in terms of the spectra of the factors and the signed coronal, including explicit results for regular and co-regular cases. It also characterizes when the product is balanced and demonstrates how to construct co-spectral signed graphs via neighbourhood corona operations, with detailed star-graph special cases yielding tractable cubic equations for eigenvalues. These results extend unsigned-corona spectral methods to signed graphs, offering new tools for graph synthesis and spectral analysis in signed networks.

Abstract

Given two signed graphs $Γ_1$ with nodes $\{u_1,u_2,\cdots,u_n\}$ and $Γ_2$, the neighbourhood corona, $Γ_1*Γ_2$ is the signed graph obtained by taking one copy of $Γ_1$ and $n_1$ copies of $Γ_2$, and joining every neighbour of the $i^{th}$ node with each nodes of the $i^{th}$ copy of $Γ_2$ by a new signed edge. In this paper we will determine the condition for $Γ_1*Γ_2$ to be balanced. We also determine the adjacency spectrum of $Γ_1*Γ_2$ for arbitrary $Γ_1$ and $Γ_2$, and Laplacian and signless Laplacian spectrum of $Γ_1*Γ_2$ for regular $Γ_1$ and arbitrary $Γ_2$, in terms of the corresponding spectrum of $Γ_1$ and $Γ_2$.

On Spectrum of Neighbourhood Corona Product of Signed Graphs

TL;DR

The paper advances the spectral theory of signed graphs by introducing and analyzing the neighbourhood corona product . It provides general closed-form relations for the adjacency, Laplacian, and signless Laplacian spectra of in terms of the spectra of the factors and the signed coronal, including explicit results for regular and co-regular cases. It also characterizes when the product is balanced and demonstrates how to construct co-spectral signed graphs via neighbourhood corona operations, with detailed star-graph special cases yielding tractable cubic equations for eigenvalues. These results extend unsigned-corona spectral methods to signed graphs, offering new tools for graph synthesis and spectral analysis in signed networks.

Abstract

Given two signed graphs with nodes and , the neighbourhood corona, is the signed graph obtained by taking one copy of and copies of , and joining every neighbour of the node with each nodes of the copy of by a new signed edge. In this paper we will determine the condition for to be balanced. We also determine the adjacency spectrum of for arbitrary and , and Laplacian and signless Laplacian spectrum of for regular and arbitrary , in terms of the corresponding spectrum of and .
Paper Structure (7 sections, 20 theorems, 51 equations, 1 figure, 2 tables)

This paper contains 7 sections, 20 theorems, 51 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Neighbourhood Corona product of signed graph
  3. Structural property of Neighbourhood corona
  4. Spectra of neighbourhood corona
  5. A-spectra of signed graph
  6. Q-spectra of signed graph
  7. L-spectra of signed graph

Key Result

Lemma 1.1

(Schur complement) bapat2010graphs Let $A$ be a $n\times n$ matrix partitioned as where $A_{11}$ and $A_{22}$ are square matrices. If $A_{11}$ and $A_{22}$ are invertible , then

Figures (1)

  • Figure 1: Neighbourhood Corona product of $\Gamma_1$ and $\Gamma_2$. The dotted line implies a negative edge and the normal line implies a positive edge.

Theorems & Definitions (37)

  • Lemma 1.1
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Proposition 2.5
  • ...and 27 more