We offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least \(\frac{n+1}{n-1}\) provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size \(\frac{n-1}{2}\). With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most \(\frac{n-1}{2}\).

中文翻译：

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归一化图拉普拉斯算子的最大特征值的谱间隙

我们提供了一种新的方法来证明具有n个顶点的图的标准化图Laplacian的最大特征值至少为\（\ frac {n + 1} {n-1} \），前提是该图不完整且相等当且仅当补图是具有两个大小均为\（\ frac {n-1} {2} \）的单边图或完整二部图时，才可实现。使用相同的方法，我们还证明了以最小顶点度为单位的最大特征值的新下限，条件是该值最多为\（\ frac {n-1} {2} \）。