Let G be a graph on n vertices and let λ₁ ⩾ λ₂ ⩾ ‣ ⩾ λ_n be the eigenvalues of its adjacency matrix. For random graphs we investigate the sum of eigenvalues \({s_k} = \sum\limits_{i = 1}^k {{\lambda _i}},\) for 1 ⩾ k ⩾ n, and show that a typical graph has S_k ⩾ (e(G) + k²)/(0.99n)^1/2, where e(G) is the number of edges of G. We also show bounds for the sum of eigenvalues within a given range in terms of the number of edges. The approach for the proofs was first used in Rocha (2020) to bound the partial sum of eigenvalues of the Laplacian matrix.

中文翻译：

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随机图特征值的部分和

让ģ是对n个顶点的曲线图，并让λ ₁ ⩾λ ₂ ⩾‣⩾λ _Ñ是其邻接矩阵的特征值。对于随机图我们研究本征值的总和\（{S_K} = \和\ limits_ {I = 1} ^ķ{{\拉姆达_i}}，\） 1⩾ ķ ⩾ Ñ，并显示一个典型的曲线图具有小号_ķ ⩾（è（ģ）+ ķ ²）/（0.99 ñ）^1/2，其中è（ģ）是边缘的数目ģ。我们还显示了给定范围内的特征值之和的边界（以边数为单位）。证明的方法首先在Rocha（2020）中用于限制Laplacian矩阵的特征值的部分和。