# Spectral Analysis of Splitting Signed Graph

## Authors

• Sandeep Kumar
• Deepa Sinha South Asian University Akbar Bhawan Chanakyapuri, New Delhi 110021 (India)

## Keywords:

Signed Graph, Splitting Signed graph, spectrum, Energy

## Abstract

An ordered pair $\Sigma = (\Sigma^{u}$,$\sigma$) is called the \textit{signed graph}, where $\Sigma^{u} = (V,E)$ is a \textit{underlying graph} and $\sigma$ is a signed mapping,
called \textit{signature}, from $E$ to the sign set $\lbrace +, - \rbrace$. The \textit{splitting signed graph} $\Gamma(\Sigma)$ of a signed graph $\Sigma$ is defined as,
for every vertex $u \in V(\Sigma)$, take a new vertex $u'$. Join $u'$ to all the vertices of $\Sigma$ adjacent to $u$ such that $\sigma_{\Gamma}(u'v) = \sigma(u'v), \ u \in N(v)$.
The objective of this paper is to propose an algorithm for the generation of a splitting signed graph, a splitting root signed graph from a given signed graph using Matlab.
Additionally, we conduct a spectral analysis of the resulting graph. Spectral analysis is performed on the adjacency and laplacian matrices of the
splitting signed graph to study its eigenvalues and eigenvectors. A relationship between the energy of the original signed graph $\Sigma$
and the energy of the splitting signed graph $\Gamma(\Sigma)$ is established.

## Author Biography

• Deepa Sinha, South Asian University Akbar Bhawan Chanakyapuri, New Delhi 110021 (India)

Department of Mathematics

Associate Professor

2024-01-31

## Section

Nonlinear Analysis

## How to Cite

Spectral Analysis of Splitting Signed Graph. (2024). European Journal of Pure and Applied Mathematics, 17(1), 504-518. https://doi.org/10.29020/nybg.ejpam.v17i1.4798

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