Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

r-orientation of a signed graph and its application on coronae of signed graphs

Satyam Guragain, Ravi Srivastava, Bishal Sonar

Abstract

For unsigned graphs G and H, the characteristic polynomial of different graph matrices for edge corona, subdivision vertex neighbourhood corona and subdivision edge neighbourhood corona has already been studied using the concept of coronal. However, till date no work regarding the spectrum of these products has been studied for signed graphs. In our work, we have filled this gap and defined these variants of coronae by introducing the concept of reverse orientation (r-orientation). We analyzed the structural properties of these product. Also, the characteristic polynomial of adjacency matrix, Laplacian matrices (signed and signless) and normalized Laplacian matrix of these variants of corona product of regular signed graphs under $r$-orientation is obtained using the concept of signed coronal. These results help us to construct infinitely many families of pairs of cospectral signed graphs.

r-orientation of a signed graph and its application on coronae of signed graphs

Abstract

For unsigned graphs G and H, the characteristic polynomial of different graph matrices for edge corona, subdivision vertex neighbourhood corona and subdivision edge neighbourhood corona has already been studied using the concept of coronal. However, till date no work regarding the spectrum of these products has been studied for signed graphs. In our work, we have filled this gap and defined these variants of coronae by introducing the concept of reverse orientation (r-orientation). We analyzed the structural properties of these product. Also, the characteristic polynomial of adjacency matrix, Laplacian matrices (signed and signless) and normalized Laplacian matrix of these variants of corona product of regular signed graphs under -orientation is obtained using the concept of signed coronal. These results help us to construct infinitely many families of pairs of cospectral signed graphs.
Paper Structure (7 sections, 56 theorems, 72 equations, 3 figures, 2 tables)

This paper contains 7 sections, 56 theorems, 72 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notations and Result used
  3. $r$-orientation of signed graphs
  4. Structural properties of $(\Gamma^1\diamond \Gamma^2)_\theta, (\Gamma^1\boxdot \Gamma^2)_\theta$ and $(\Gamma^1\boxminus \Gamma^2)_\theta$
  5. Spectrum and Laplacian spectrum of $(\Gamma^1\diamond \Gamma^2)_\theta$
  6. Spectrum and Laplacian spectrum of $(\Gamma^1\boxdot \Gamma^2)_\theta$ and $(\Gamma^1\boxminus \Gamma^2)_\theta$
  7. Normalized Laplacian spectrum of $(\Gamma^1\diamond\Gamma^2)_\theta$, $(\Gamma^1\boxdot\Gamma^2)_\theta$ and $(\Gamma^1\boxminus\Gamma^2)_\theta$

Key Result

Lemma 2.3

Let $\Gamma=(G,\sigma)$ be a signed graph on $n$ vertices and $m$ edges and $\theta$ be any $r$-orientation of edges of $\Gamma$ then $R(\Gamma_\theta)R(\Gamma_\theta)^T=Q(\Gamma)$.

Figures (3)

  • Figure 1: orientation and $r$-orientation of (a) positive and (b) negative edges.
  • Figure 2: A line signed graph $\mathcal{L}(\Gamma_\theta)$ and a subdivision signed graph $S(\Gamma_\theta)$ of a signed graph $\Gamma$ under $r$-orientation $\theta$ of edges of $\Gamma$.
  • Figure 3: Subdivision graph, edge corona, subdivision vertex neighbourhood and subdivision edge neighbourhood corona of signed graphs under $r$-orientation.

Theorems & Definitions (86)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Remark 2.7
  • ...and 76 more