Abstract
An ordered pair Σ = (Σu, σ) is called the signed graph, where Σu = (V, E) is an underlying graph and σ is a signed mapping, called signature, from E to the sign set { + , - }. A marking of Σ is a function μ: V(Σ) → { + , - }. The canonical marking of a signed graph Σ, denoted μσ, is given as μσ(v) = Πvu∈E(Σ)σ(vu). The canonical splitting signed graphξ(Σ) of a signed graph Σ is defined as a signed graph ξ(Σ) = (V(ξ), E(ξ)) , with V(ξ) = V(Σ) ∪ V', where V' is copy of a vertex set in V(Σ) s.t. for each vertex u ∈ V(Σ), take a new vertex u' and E(ξ) is defined as, join u' to all the vertices of Σ adjacent to u by negative edge if μσ(u) = μσ(v) = - , where v ∈ N(u) and by positive edge otherwise. The objective of this paper is to propose an algorithm for the generation of a canonical splitting signed graph, a splitting root signed graph from a given signed graph, provided it exists and to give the characterization of balanced canonical splitting signed graph. Additionally, we conduct a spectral analysis of the resulting graph. Spectral analysis is performed on the adjacency and Laplacian matrices of the canonical 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 canonical splitting signed graph ξ(Σ) is established. •Algorithm to generate canonical splitting signed graph ξ(Σ).•Spectral Analysis is performed for both adjacency and Laplacian matrices of canonical splitting signed graph ξ(Σ).