Vertex Partitioning and $p$-Energy of Graphs

Abstract: 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), \quad \mathcal{E}p^-(A)=\sum{\lambda_i<0} |\lambda_i(A)|^p,] to be the positive and the negative $p$-energy of $A$, respectively. In this note, first we show that if $A=[A_{ij}]{i,j=1}^k$, where $A{ii}$ are square matrices, then [ \mathcal{E}p^+(A)\geq \sum{i=1}^{k} \mathcal{E}p^+(A{ii}), \quad \mathcal{E}p^-(A)\geq \sum{i=1}^{k} \mathcal{E}p^-(A{ii}),] for any real number $p\geq 1$. We then apply the previous inequality to establish lower bounds for $p$-energy of the adjacency matrix of graphs.