Spectral Analysis of Splitting Signed Graph (European Journal of Pure and Applied Mathematics, 2024)
An ordered pair $Σ = (Σ^{u}$,$σ$) is called the signed graph, where $Σ^{u} = (V,E)$ is a underlying graph and $σ$ is a signed mapping, called signature, from $E$ to the sign set $\lbrace +, - \rbrace$. The splitting signed graph $Γ(Σ)$ of a signed graph $Σ$ is defined as, for every vertex $u \in V(Σ)$, take a new vertex $u'$. Join $u'$ to all the vertices of $Σ$ adjacent to $u$ such that $σ_{Γ}(u'v) = σ(u'v), \ u \in N(v)$. 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 $Σ$ and the energy of the splitting signed graph $Γ(Σ)$ is established.
Keywords: Signed Graph; Splitting Signed graph; spectrum; Energy