For a simple graph G with order n and size m having Laplacian eigenvalues \(\mu _1, \mu _2, \dots , \mu _{n-1},\mu _n=0\), let \(S_k(G)=\sum _{i=1}^{k}\mu _i\), be the sum of k largest Laplacian eigenvalues of G. We obtain upper bounds for the sum of k largest Laplacian eigenvalues of two large families of graphs. As a consequence, we prove Brouwer’s Conjecture for large number of graphs which belong to these families of graphs.

中文翻译：

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图和集团数的k个最大拉普拉斯特征值之和

对于具有n阶和大小为m的具有拉普拉斯特征值\（\ mu _1，\ mu _2，\ dots，\ mu _ {n-1}，\ mu _n = 0 \）的简单图G，令\（S_k（G ）= \ sum _ {i = 1} ^ {k} \ mu _i \）是G的k个最大拉普拉斯特征值之和。我们获得了两个大图族的k个最大拉普拉斯特征值之和的上限。结果，我们证明了属于这些图族的大量图的布劳威尔猜想。