Permanental Energy of Graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913064733900800 |
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| author | Pant, Priyanshu Singh, Ranveer |
| author_facet | Pant, Priyanshu Singh, Ranveer |
| contents | For a simple graph $G$ with adjacency matrix $A(G)$, let $π(G,x):=\mathrm{per}(xI-A(G))$ be its permanental polynomial with roots $μ_1,\ldots,μ_n \in \mathbb{C}$, and define the permanental energy $E_{\mathrm{per}}(G):=\sum_{i=1}^n |μ_i|$. We prove a sharp universal lower bound: for every $m$-edge graph $G$, $E_{\mathrm{per}}(G) \ge 2\sqrt{m}$, with equality if and only if $G$ is a star together with isolated vertices. We also prove the general upper bound $E_{\mathrm{per}}(G) \le nρ(G)$, where $ρ(G)$ is the spectral radius, and we study $E_{\mathrm{per}}(G)$ on several graph families. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_24165 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Permanental Energy of Graphs Pant, Priyanshu Singh, Ranveer Combinatorics Discrete Mathematics Spectral Theory 05C50, 05C35, 15A15, 15A18 G.2.2; F.2.2 For a simple graph $G$ with adjacency matrix $A(G)$, let $π(G,x):=\mathrm{per}(xI-A(G))$ be its permanental polynomial with roots $μ_1,\ldots,μ_n \in \mathbb{C}$, and define the permanental energy $E_{\mathrm{per}}(G):=\sum_{i=1}^n |μ_i|$. We prove a sharp universal lower bound: for every $m$-edge graph $G$, $E_{\mathrm{per}}(G) \ge 2\sqrt{m}$, with equality if and only if $G$ is a star together with isolated vertices. We also prove the general upper bound $E_{\mathrm{per}}(G) \le nρ(G)$, where $ρ(G)$ is the spectral radius, and we study $E_{\mathrm{per}}(G)$ on several graph families. |
| title | Permanental Energy of Graphs |
| topic | Combinatorics Discrete Mathematics Spectral Theory 05C50, 05C35, 15A15, 15A18 G.2.2; F.2.2 |
| url | https://arxiv.org/abs/2604.24165 |