Vertex Partitioning and $p$-Energy of Graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909654432350208 |
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| author | Akbari, Saieed Kumar, Hitesh Mohar, Bojan Pragada, Shivaramakrishna |
| author_facet | Akbari, Saieed Kumar, Hitesh Mohar, Bojan Pragada, Shivaramakrishna |
| contents | For a Hermitian matrix $A$ of order $n$ with eigenvalues $λ_1(A)\ge \cdots\ge λ_n(A)$, 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 $p\geq 1$. We then apply the previous inequality to establish lower bounds for $p$-energy of the adjacency matrix of graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16882 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vertex Partitioning and $p$-Energy of Graphs Akbari, Saieed Kumar, Hitesh Mohar, Bojan Pragada, Shivaramakrishna Combinatorics For a Hermitian matrix $A$ of order $n$ with eigenvalues $λ_1(A)\ge \cdots\ge λ_n(A)$, 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 $p\geq 1$. We then apply the previous inequality to establish lower bounds for $p$-energy of the adjacency matrix of graphs. |
| title | Vertex Partitioning and $p$-Energy of Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.16882 |