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On spectrum of corona product of duplication signed graph and its application

Bishal Sonar, Ravi Srivastava

TL;DR

This work introduces μ-signed and duplication signed graphs and proves they are structurally balanced, then develops two corona products based on duplication, showing they are switching isomorphic. It derives comprehensive adjacency, Laplacian, and signless Laplacian spectra for these products in terms of the base graphs’ spectra, with explicit factorized characteristic polynomials using the signed coronal. The authors analyze structural properties, balance criteria, and co-spectral implications, and demonstrate applications to generate integral signed graphs and non-co-spectral equienergetic families. Overall, the paper provides a unified spectral framework for duplication-based corona products and offers practical methods for constructing specialized signed graphs with desired spectral features.

Abstract

This paper introduces the concept of $μ$-signed and duplication signed graphs and shows that both are always structurally balanced. Using the duplication signed graph, we define the corona product of the duplication signed graph (Duplication add vertex corona product and duplication vertex corona product) and explore their structural properties. Additionally, we provide the adjacency spectrum of both the products for any $Γ_1$ and $Γ_2$, and the Laplacian and signless Laplacian spectrum for regular $Γ_1$ and arbitrary $Γ_2$, in terms of the corresponding spectrum of $Γ_1$ and $Γ_2$. Finally, we discuss its application in generating integral and equienergetic signed graphs.

On spectrum of corona product of duplication signed graph and its application

TL;DR

This work introduces μ-signed and duplication signed graphs and proves they are structurally balanced, then develops two corona products based on duplication, showing they are switching isomorphic. It derives comprehensive adjacency, Laplacian, and signless Laplacian spectra for these products in terms of the base graphs’ spectra, with explicit factorized characteristic polynomials using the signed coronal. The authors analyze structural properties, balance criteria, and co-spectral implications, and demonstrate applications to generate integral signed graphs and non-co-spectral equienergetic families. Overall, the paper provides a unified spectral framework for duplication-based corona products and offers practical methods for constructing specialized signed graphs with desired spectral features.

Abstract

This paper introduces the concept of -signed and duplication signed graphs and shows that both are always structurally balanced. Using the duplication signed graph, we define the corona product of the duplication signed graph (Duplication add vertex corona product and duplication vertex corona product) and explore their structural properties. Additionally, we provide the adjacency spectrum of both the products for any and , and the Laplacian and signless Laplacian spectrum for regular and arbitrary , in terms of the corresponding spectrum of and . Finally, we discuss its application in generating integral and equienergetic signed graphs.
Paper Structure (11 sections, 20 theorems, 43 equations, 2 figures, 2 tables)

This paper contains 11 sections, 20 theorems, 43 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Corona product of duplication signed graph
  4. Structural properties
  5. Spectral properties
  6. A-spectra of signed graph
  7. L-spectra of signed graph
  8. Application
  9. Families of Integral Signed Graphs
  10. Equienergetic signed graphs
  11. Conclusion

Key Result

Theorem 2.1

harary1953notion A signed graph $\Gamma=(G,\sigma,\mu)$ is balanced if and only if there exists a marking $\mu$ of its vertices such that for each edge $uv$ in $G$ one has $\sigma(uv)=\mu(u)\mu(v).$

Figures (2)

  • Figure 1: $A=P_2\circledast C_3$ and $B=P_2\circledcirc C_3$. The normal line implies a positive edge, and the broken line implies a negative edge.
  • Figure 2: $P_2$ is balanced but both $P_3\circledast P_2$ and $P_3\circledcirc P_2$ are unbalanced.

Theorems & Definitions (40)

  • Definition 2.1: Duplication signed graph
  • Definition 2.2: $\mu$-signed graph
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Definition 2.3
  • ...and 30 more