Some Upper Bounds for the Net Laplacian Index of a Signed Graph

Abstract

The net Laplacian matrix NG˙ of a signed graph G˙ is defined as NG˙=DG˙±-AG˙, where DG˙± and AG˙ denote the diagonal matrix of net-degrees and the adjacency matrix of G˙ , respectively. In this study, we give two upper bounds for the largest eigenvalue of NG˙, both expressed in terms related to vertex degrees. We also discuss their quality, provide certain comparisons and consider some particular cases.