The maximum number of connected sets in regular graphs

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, Ndom(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 Ndom(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 Mvertices and the same regularity. We improve the best known upper bounds for N(G) and Ndom(G) conditional on this conjecture.