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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.14686 |
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| _version_ | 1866918450490769408 |
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| author | Dede, Cahit Popat, Kalpesh M. |
| author_facet | Dede, Cahit Popat, Kalpesh M. |
| contents | For a given graph \( G \), let \( G^{(j)} \) denote the graph obtained by the deletion of vertex \( v_j \) from \( G \). The difference \( \mathscr{E}(G) - \mathscr{E}(G^{(j)}) \) quantifies the change in the energy of \( G \) upon the removal of \( v_j \), termed as the local energy of \( G \) at vertex $v_j$, as defined by Espinal and Rada in 2024. The local energy of $G$ at vertex $v$ is denoted by \(\mathscr{E}_G(v)\). The local energy of the graph \( G \), therefore, is the summation of these vertex-specific local energies across all vertices in \( V(G) \), expressed by \( e(G) = \sum \mathscr{E}_G(v) \). Two graphs of the same order are defined as locally equienergetic if they have identical local energy. In this paper, we have investigated several pairs of locally equienergetic graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14686 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Locally Equienergetic Graphs Dede, Cahit Popat, Kalpesh M. Combinatorics Spectral Theory For a given graph \( G \), let \( G^{(j)} \) denote the graph obtained by the deletion of vertex \( v_j \) from \( G \). The difference \( \mathscr{E}(G) - \mathscr{E}(G^{(j)}) \) quantifies the change in the energy of \( G \) upon the removal of \( v_j \), termed as the local energy of \( G \) at vertex $v_j$, as defined by Espinal and Rada in 2024. The local energy of $G$ at vertex $v$ is denoted by \(\mathscr{E}_G(v)\). The local energy of the graph \( G \), therefore, is the summation of these vertex-specific local energies across all vertices in \( V(G) \), expressed by \( e(G) = \sum \mathscr{E}_G(v) \). Two graphs of the same order are defined as locally equienergetic if they have identical local energy. In this paper, we have investigated several pairs of locally equienergetic graphs. |
| title | Locally Equienergetic Graphs |
| topic | Combinatorics Spectral Theory |
| url | https://arxiv.org/abs/2604.14686 |