Monodromy representation of graphs
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914051034972160 |
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| author | Yuan, Kai Wang, Yan |
| author_facet | Yuan, Kai Wang, Yan |
| contents | It is well-known that every vertex-transitive graph admits a representation as a coset graph. In this paper, we extend this construction by introducing monodromy graphs defined through double cosets. Our main result establishes that every graph is isomorphic to a monodromy graph, providing a new combinatorial framework for graph representation. Moreover, we show that every graph gives rise to an arc-transitive graph through its monodromy representation. Inspired by the monodromy representation of graphs, we denote an algebraic map $\mathcal{M}(G;Ω,ρ,τ)$ by $\mathcal{M}(G;U,ρ,τ)$ where $U$ is a stabiliser in $G$. As an application, we prove an enumeration theorem for orientable maps with a given monodromy group. We underscore a fundamental triad in algebraic graph theory: Where there is a graph, there is a group, an arc-transitive graph, and an orientable regular map--each arising canonically from the underlying combinatorial and algebraic structures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17910 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Monodromy representation of graphs Yuan, Kai Wang, Yan Combinatorics Group Theory 05C25, 05C30, 05E16, 05C62 It is well-known that every vertex-transitive graph admits a representation as a coset graph. In this paper, we extend this construction by introducing monodromy graphs defined through double cosets. Our main result establishes that every graph is isomorphic to a monodromy graph, providing a new combinatorial framework for graph representation. Moreover, we show that every graph gives rise to an arc-transitive graph through its monodromy representation. Inspired by the monodromy representation of graphs, we denote an algebraic map $\mathcal{M}(G;Ω,ρ,τ)$ by $\mathcal{M}(G;U,ρ,τ)$ where $U$ is a stabiliser in $G$. As an application, we prove an enumeration theorem for orientable maps with a given monodromy group. We underscore a fundamental triad in algebraic graph theory: Where there is a graph, there is a group, an arc-transitive graph, and an orientable regular map--each arising canonically from the underlying combinatorial and algebraic structures. |
| title | Monodromy representation of graphs |
| topic | Combinatorics Group Theory 05C25, 05C30, 05E16, 05C62 |
| url | https://arxiv.org/abs/2509.17910 |