Some Properties of the Eigenvalues of the Net Laplacian Matrix of a Signed Graph

Abstract

Given a signed graph Ġ, let AĠ and DG˙±D_{\dot G}^ \pm denote its standard adjacency matrix and the diagonal matrix of vertex net-degrees, respectively. The net Laplacian matrix of Ġ is defined to be NG˙=DG˙±-AG˙{N_{\dot G}} = D_{\dot G}^ \pm - {A_{\dot G}} . In this study we give some properties of the eigenvalues of NĠ. In particular, we consider their behaviour under some edge perturbations, establish some relations between them and the eigenvalues of the standard Laplacian matrix and give some lower and upper bounds for the largest eigenvalue of NĠ.

Keywords

• (net) laplacian matrix
• edge perturbations
• largest eigenvalue
• net-degree
• 05c22
• 05c50