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 λ1≥λ2≥⋯≥λn 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=1kλi≤n2(θ+1)(θ+θ(kθ+k−1)) with equality holding if and only if G≅nK1 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=2n2(4d−n)2, where d>n4. Finally we give a partial solution to this problem.