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.

中文翻译：

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签名图的净拉普拉斯指数的一些上限

有符号图\（\ dot {G} \）的净拉普拉斯矩阵\（N _ {\ dot {G}} \）定义为\（N _ {\ dot {G}} = D _ {\ dot {G} } ^ {\ pm} -A _ {\ dot {G}} \），其中\（D _ {\ dot {G}} ^ {\ pm} \）和\（A _ {\ dot {G}} \）表示净度的对角矩阵和\（\ dot {G} \）的邻接矩阵。在这项研究中，我们给出了\（N _ {\ dot {G}} \）的最大特征值的两个上限，两者均以与顶点度有关的术语表示。我们还将讨论它们的质量，提供一些比较并考虑一些特殊情况。