  We make some fundamental observations and conjectures on the number of
connected sets $N(G)$ in $d$-regular graphs $G$. We improve the best known
lower bounds on the exponential behavior of the maximum of $N(G)$ for regular
graphs by considering a different construction of a family of graphs (depending
on smaller base graphs) and improve the upper bounds conditional on one of our
conjectures. The lower bounds are estimated using combinatorial reductions and
linear algebra. We also determine the exact maxima of $N(G)$ for cubic and
quartic graphs with small order.
