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 \[ \mathcal{E}_p^+(A)=\sum_{λ_i > 0} λ_i^p(A), \quad \mathcal{E}_p^-(A)=\sum_{λ_i<0} |λ_i(A)|^p,\] to be the positive and the negative $p$-energy of $A$, respectively. In this note, first we show that if $A=[A_{ij}{]}_{i,j=1}^k$, 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 $p\ge 1$. We then apply the previous inequality to establish lower bounds for $p$-energy of the adjacency matrix of graphs.