Vertex Partitioning and $p$-Energy of Graphs

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