Vertex partitioning and $p$-energy of graphs

Akbari, Saieed (author), Kumar, Hitesh (author), Mohar, Bojan (author), Pragada, Shivaramakrishna (author)

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.

• Language: English
• Keywords: adjacency matrix, graph energy, positive p-energy, negative p-energy, Schatten p-norm, vertex partition
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.11.2025
• Year: 2025
• Number of pages: Str. 96-107
• Numbering: Vol. 724
• PID: 20.500.12556/RUL-175449
• UDC: 519.17
• ISSN on article: 0024-3795
• DOI: 10.1016/j.laa.2025.06.009
• COBISS.SI-ID: 241730563
• Publication date in RUL: 28.10.2025

Record is a part of a journal

• Title: Linear algebra and its applications
• Shortened title: Linear algebra appl.
• Publisher: Elsevier
• ISSN: 0024-3795
• COBISS.SI-ID: 1119247

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