The maximum number of connected sets in regular graphs
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909326691532800 |
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| author | Cambie, Stijn Goedgebeur, Jan Jooken, Jorik |
| author_facet | Cambie, Stijn Goedgebeur, Jan Jooken, Jorik |
| contents | We improve the best known lower bounds on the exponential behavior of the maximum of the number of connected sets, $N(G)$, and dominating connected sets, $N_{dom}(G)$, for regular graphs. These lower bounds are improved by constructing a family of graphs defined in terms of a small base graph (a Moore graph), using a combinatorial reduction of these graphs to rectangular boards followed by using linear algebra to show that the lower bound is related to the largest eigenvalue of a coefficient matrix associated with the base graph. We also determine the exact maxima of $N(G)$ and $N_{dom}(G)$ for cubic and quartic graphs of small order. We give multiple results in favor of a conjecture that each Moore graph $M$ maximizes the base indicating the exponential behavior of the number of connected vertex subsets among graphs with at least $|M|$ vertices and the same regularity. We improve the best known upper bounds for $N(G)$ and $N_{dom}(G)$ conditional on this conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_00075 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The maximum number of connected sets in regular graphs Cambie, Stijn Goedgebeur, Jan Jooken, Jorik Combinatorics 05C07, 05C35, 05C40, 05C48, 05C50, 05C69, 05C85, 68R05, 68R10 We improve the best known lower bounds on the exponential behavior of the maximum of the number of connected sets, $N(G)$, and dominating connected sets, $N_{dom}(G)$, for regular graphs. These lower bounds are improved by constructing a family of graphs defined in terms of a small base graph (a Moore graph), using a combinatorial reduction of these graphs to rectangular boards followed by using linear algebra to show that the lower bound is related to the largest eigenvalue of a coefficient matrix associated with the base graph. We also determine the exact maxima of $N(G)$ and $N_{dom}(G)$ for cubic and quartic graphs of small order. We give multiple results in favor of a conjecture that each Moore graph $M$ maximizes the base indicating the exponential behavior of the number of connected vertex subsets among graphs with at least $|M|$ vertices and the same regularity. We improve the best known upper bounds for $N(G)$ and $N_{dom}(G)$ conditional on this conjecture. |
| title | The maximum number of connected sets in regular graphs |
| topic | Combinatorics 05C07, 05C35, 05C40, 05C48, 05C50, 05C69, 05C85, 68R05, 68R10 |
| url | https://arxiv.org/abs/2311.00075 |