Spectral Analysis of Splitting Signed Graph

Abstract

An ordered pair $\mathrm{\Sigma }=({\mathrm{\Sigma }}^u$,$\sigma$) is called the \textit{signed graph}, where ${\mathrm{\Sigma }}^u=(V,E)$ is a \textit{underlying graph} and $\sigma$ is a signed mapping,
called \textit{signature}, from $E$ to the sign set $\{+,-\}$. The \textit{splitting signed graph} $\mathrm{\Gamma }(\mathrm{\Sigma })$ of a signed graph $\mathrm{\Sigma }$ is defined as,
for every vertex $u\in V(\mathrm{\Sigma })$, take a new vertex $u^{\prime }$. Join $u^{\prime }$ to all the vertices of $\mathrm{\Sigma }$ adjacent to $u$ such that ${\sigma }_{\mathrm{\Gamma }}(u^{\prime }v)=\sigma (u^{\prime }v),u\in N(v)$.
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 $\mathrm{\Sigma }$
and the energy of the splitting signed graph $\mathrm{\Gamma }(\mathrm{\Sigma })$ is established.