Abstract The spectrum of the normalized Laplacian matrix of a graph provides many structural information of the graph, and it has many applications in numerous areas and in different guises. Let G be a complete k -partite graph with k ≥ 3 . In this paper, we give the necessary and sufficient condition for G which is determined by their normalized Laplacian spectrum. Moreover, we obtain a majorization theory of normalized Laplacian spectral radius of G , which enables us to find the maximal and minimal normalized Laplacian spectral radii among all complete k -partite graphs with fixed order, respectively.

中文翻译：

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完整多部分图的归一化拉普拉斯谱

摘要 图的归一化拉普拉斯矩阵的谱提供了图的许多结构信息，它在众多领域和不同形式中有许多应用。令 G 是一个完整的 k 部分图，其中 k ≥ 3 。在本文中，我们给出了由它们的归一化拉普拉斯谱确定的 G 的充要条件。此外，我们获得了 G 的归一化拉普拉斯谱半径的主化理论，这使我们能够分别在所有具有固定阶数的完整 k 部分图中找到最大和最小归一化拉普拉斯谱半径。