Spectral Analysis of Splitting Signed Graph

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.