In this paper, we consider the spectrum of a Laplacian matrix, also known as Markov matrices where the entries of the matrix are independent but have a variance profile. Motivated by recent works on generalized Wigner matrices we assume that the variance profile gives rise to a sequence of graphons. Under the assumption that these graphons converge, we show that the limiting spectral distribution converges. We give an expression for the moments of the limiting measure in terms of graph homomorphisms. In some special cases, we identify the limit explicitly. We also study the spectral norm and derive the order of the maximum eigenvalue. We show that our results cover Laplacians of various random graphs including inhomogeneous Erdős–Rényi random graphs, sparse W-random graphs, stochastic block matrices and constrained random graphs.

中文翻译：

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广义 Wigner 矩阵的拉普拉斯算子的谱性质

在本文中，我们考虑拉普拉斯矩阵的谱，也称为马尔可夫矩阵，其中矩阵的条目是独立的，但具有方差分布。受最近关于广义 Wigner 矩阵的研究的启发，我们假设方差分布会产生一系列石墨子。在这些石墨子收敛的假设下，我们表明极限光谱分布收敛。我们根据图同态给出了极限测度矩的表达式。在某些特殊情况下，我们会明确确定限制。我们还研究了谱范数并推导出最大特征值的阶数。我们表明我们的结果涵盖了各种随机图的拉普拉斯算子，包括非齐次 Erdős-Rényi 随机图、稀疏W- 随机图、随机块矩阵和约束随机图。