The Pentagon Graph Operator
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910154602053632 |
|---|---|
| author | Gervacio, Severino V. Maehara, Hiroshi Ramos, Phoebe Chloe |
| author_facet | Gervacio, Severino V. Maehara, Hiroshi Ramos, Phoebe Chloe |
| contents | For a graph $G$, let $\mathscr{C}_5(G)$ denote the graph whose vertices are the induced $5$-cycles of $G$, where two vertices are adjacent whenever the corresponding cycles share an edge. We investigate the iterative behavior of the pentagon graph operator $\mathscr{C}_5(G)$ , positioning it as the natural continuation of the quadrangle graph operator and the broader induced-cycle graph operator program. We construct explicit pentagon-vanishing, pentagon-periodic, and pentagon-expanding graphs. In particular, the dodecahedron and the icosahedron provide natural periodic examples, while an icosahedral tadpole-hat construction yields expanding families. Our main result proves that every graph is exactly one of three types with respect to $\mathscr{C}_5(G)$: vanishing, periodic, or expanding. The paper suggests a broader theory for the operators $C_k$ generated by induced cycles of fixed length $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18984 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Pentagon Graph Operator Gervacio, Severino V. Maehara, Hiroshi Ramos, Phoebe Chloe Combinatorics 05C62 For a graph $G$, let $\mathscr{C}_5(G)$ denote the graph whose vertices are the induced $5$-cycles of $G$, where two vertices are adjacent whenever the corresponding cycles share an edge. We investigate the iterative behavior of the pentagon graph operator $\mathscr{C}_5(G)$ , positioning it as the natural continuation of the quadrangle graph operator and the broader induced-cycle graph operator program. We construct explicit pentagon-vanishing, pentagon-periodic, and pentagon-expanding graphs. In particular, the dodecahedron and the icosahedron provide natural periodic examples, while an icosahedral tadpole-hat construction yields expanding families. Our main result proves that every graph is exactly one of three types with respect to $\mathscr{C}_5(G)$: vanishing, periodic, or expanding. The paper suggests a broader theory for the operators $C_k$ generated by induced cycles of fixed length $k$. |
| title | The Pentagon Graph Operator |
| topic | Combinatorics 05C62 |
| url | https://arxiv.org/abs/2604.18984 |