On a magneto-spectral invariant on finite graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917170950176768 |
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| author | Hu, Chunyang Hua, Bobo Kamtue, Supanat Liu, Shiping Münch, Florentin Peyerimhoff, Norbert |
| author_facet | Hu, Chunyang Hua, Bobo Kamtue, Supanat Liu, Shiping Münch, Florentin Peyerimhoff, Norbert |
| contents | In this paper, we introduce a magneto-spectral invariant for finite graphs. This invariant vanishes on trees and is maximized by complete graphs. We compute this invariant for cycles, complete graphs, wheel graphs, hypercubes, complete bipartite graphs and suspensions of trees and derive various lower and upper bounds. In particular, we provide a sharp upper bound for regular bipartite graphs and derive a direct relation between the class of graphs assuming this upper bound and the class of unit weighing matrices, which are generalizations of complex Hadamard matrices. Moreover, this class of bipartite graphs has non-negative magnetic Bakry-Émery curvature and is preserved under both the Cartesian product and a partial tensor product for bipartite graphs. The study of our invariant for certain pairs of cospectral graphs indicates also that this invariant allows us to distinguish between them. Finally, we discuss the behaviour of this invariant under various graph operations and investigate relations to the spectral gap. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03754 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a magneto-spectral invariant on finite graphs Hu, Chunyang Hua, Bobo Kamtue, Supanat Liu, Shiping Münch, Florentin Peyerimhoff, Norbert Spectral Theory Combinatorics 05C50 (Primary) 15A18, 05C76 (Secondary) In this paper, we introduce a magneto-spectral invariant for finite graphs. This invariant vanishes on trees and is maximized by complete graphs. We compute this invariant for cycles, complete graphs, wheel graphs, hypercubes, complete bipartite graphs and suspensions of trees and derive various lower and upper bounds. In particular, we provide a sharp upper bound for regular bipartite graphs and derive a direct relation between the class of graphs assuming this upper bound and the class of unit weighing matrices, which are generalizations of complex Hadamard matrices. Moreover, this class of bipartite graphs has non-negative magnetic Bakry-Émery curvature and is preserved under both the Cartesian product and a partial tensor product for bipartite graphs. The study of our invariant for certain pairs of cospectral graphs indicates also that this invariant allows us to distinguish between them. Finally, we discuss the behaviour of this invariant under various graph operations and investigate relations to the spectral gap. |
| title | On a magneto-spectral invariant on finite graphs |
| topic | Spectral Theory Combinatorics 05C50 (Primary) 15A18, 05C76 (Secondary) |
| url | https://arxiv.org/abs/2507.03754 |