Partial Sum of Eigenvalues of Random Graphs

Abstract

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.