On net-Laplacian Energy of Signed Graphs

Abstract

A signed graph is a graph where the edges are assigned either positive ornegative signs. Net degree of a signed graph is the di erence between the number ofpositive and negative edges incident with a vertex. It is said to be net-regular if all itsvertices have the same net-degree. Laplacian energy of a signed graph is defi ned asε(L( Σ)) =|γ_1-(2m)/n|+...+|γ_n-(2m)/n| where γ_1,...,γ_n are the eigenvalues of L(Σ ) and (2m)/n isthe average degree of the vertices in Σ . In this paper, we de ne net-Laplacian matrixconsidering the edge signs of a signed graph and give bounds for signed net-Laplacianeigenvalues. Further, we introduce net-Laplacian energy of a signed graph and establishnet-Laplacian energy bounds.