Vertex partitioning and p-energy of graphs

For a Hermitian matrix A of order n with eigenvalues λ 1 ( A ) ≥ ⋯ ≥ λ n ( A ) , define E p + ( A ) = ∑ λ i > 0 λ i p ( A ) , E p − ( A ) = ∑ λ i < 0 | λ i ( A ) | p , to be the positive and the negative p -energy of A , respectively. In this note, first we show that if A = [ A i j ] i , j = 1 k , where A i i are square matrices, then E p + ( A ) ≥ ∑ i = 1 k E p + ( A i i ) , E p − ( A ) ≥ ∑ i = 1 k E p − ( A i i ) , for any real number p ≥ 1 . We then apply the previous inequalities to establish lower bounds for p -energy of the adjacency matrix of graphs.