On the sum of the k largest eigenvalues of graphs and maximal energy of bipartite graphs

Let G be a graph of order n. Also let λ ₁ ≥λ ₂ ≥⋯≥λ _n be the eigenvalues of graph G. In this paper, we present the following upper bound on the sum of the k (≤n) largest eigenvalues of G in terms of the order n and negative inertia θ (the number of negative eigenvalues): ∑i=1kλ _i ≤[Formula presented](θ+θ(kθ+k−1)) with equality holding if and only if G≅nK ₁ or G≅pK‾ _t (n=pt,p≥2) with k=1 (where pK‾ _t is the complete p-partite graph of p t vertices with all partition sets having size t). Moreover, we obtain a sharp upper bound on the energy of bipartite graphs with given order n and rank r. We also propose an open problem as follows: Characterize all connected d-regular bipartite graphs of order n with 5 distinct eigenvalues and rank r=[Formula presented], where d>[Formula presented]. Finally we give a partial solution to this problem.