On the spectrum and energy of digraphs with self-loops
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915442183897088 |
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| author | Fung, Kevin Lim, Johnny |
| author_facet | Fung, Kevin Lim, Johnny |
| contents | A digraph with self-loops $D_S$ with vertex set $\mathcal{V}$ is a simple digraph with a self-loop attached at every vertex in $S \subset \mathcal{V}.$ In this paper, we study the energy $E(D_S)$ of $D_S$ and its properties, which extend several classical results on simple directed graphs. If $D_1,...,D_k$ are the strong components of $D_S,$ we establish a necessary and sufficient conditions for $E(D_S) \leq \sum^k_{i=1} E(D_i),$ for which the strict inequality exists for $S\neq \emptyset.$ We also provide several bounds and characterizations for the energy and spectral radius of $D_S,$ including the McClelland type bound. Lastly, we propose a notion of the complement of $D_S$ and establish some formulae describing the relationship between the energy and spectrum of regular digraphs with their complement. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07004 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the spectrum and energy of digraphs with self-loops Fung, Kevin Lim, Johnny Combinatorics 05C50, 05C90, 05C92 A digraph with self-loops $D_S$ with vertex set $\mathcal{V}$ is a simple digraph with a self-loop attached at every vertex in $S \subset \mathcal{V}.$ In this paper, we study the energy $E(D_S)$ of $D_S$ and its properties, which extend several classical results on simple directed graphs. If $D_1,...,D_k$ are the strong components of $D_S,$ we establish a necessary and sufficient conditions for $E(D_S) \leq \sum^k_{i=1} E(D_i),$ for which the strict inequality exists for $S\neq \emptyset.$ We also provide several bounds and characterizations for the energy and spectral radius of $D_S,$ including the McClelland type bound. Lastly, we propose a notion of the complement of $D_S$ and establish some formulae describing the relationship between the energy and spectrum of regular digraphs with their complement. |
| title | On the spectrum and energy of digraphs with self-loops |
| topic | Combinatorics 05C50, 05C90, 05C92 |
| url | https://arxiv.org/abs/2508.07004 |