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

Abstract

The net Laplacian matrix $N_{\dot{G}}$ of a signed graph $\dot{G}$ is defined as $N_{\dot{G}}=D_{\dot{G}}^{\pm }-A_{\dot{G}}$, where $D_{\dot{G}}^{\pm }$ and $A_{\dot{G}}$ denote the diagonal matrix of net-degrees and the adjacency matrix of $\dot{G}$, respectively. In this study, we give two upper bounds for the largest eigenvalue of $N_{\dot{G}}$, both expressed in terms related to vertex degrees. We also discuss their quality, provide certain comparisons and consider some particular cases.