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arXiv · 2311.00075

The maximum number of connected sets in regular graphs

Stijn Cambie·Jan Goedgebeur·Jorik Jooken
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Abstract

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.

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Stijn Cambie, Jan Goedgebeur, Jorik Jooken. 2024-09-26. The maximum number of connected sets in regular graphs. https://arxiv.org/abs/2311.00075

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