Vertex partitioning and p-energy of graphs

Abstract

For a Hermitian matrix A of order n with eigenvalues ${\lambda }_1\left(A\right)\ge \cdots \ge {\lambda }_n\left(A\right)$, define$E_p^{+}\left(A\right)=\sum _{{\lambda }_i>0}{\lambda }_i^p\left(A\right),E_p^{-}\left(A\right)=\sum _{{\lambda }_i<0}\left|{\lambda }_i\right(A){|}^p,$ to be the positive and the negative p-energy of A, respectively. In this note, first we show that if $A={\left[A_{ij}\right]}_{i,j=1}^k$, where $A_{ii}$ are square matrices, then$E_p^{+}\left(A\right)\ge \sum _{i=1}^k{E}_p^{+}\left(A_{ii}\right),E_p^{-}\left(A\right)\ge \sum _{i=1}^k{E}_p^{-}\left(A_{ii}\right),$ for any real number $p\ge 1$. We then apply the previous inequalities to establish lower bounds for p-energy of the adjacency matrix of graphs.