  In this paper we consider the number of Hamilton cycles in planar cubic
graphs of high cyclic edge-connectivity, answering two questions raised by Chia
and Thomassen ("On the number of longest and almost longest cycles in cubic
graphs", Ars Combin., 104, 307--320, 2012) about extremal graphs in these
families. In particular, we find families of cyclically $5$-edge connected
planar cubic graphs with more Hamilton cycles than the generalized Petersen
graphs $P(2n,2)$. The graphs themselves are fullerene graphs that correspond to
certain carbon molecules known as nanotubes --- more precisely, the family
consists of the zigzag nanotubes of (fixed) width $5$ and increasing length. In
order to count the Hamilton cycles in the nanotubes, we develop methods
inspired by the transfer matrices of statistical physics. We outline how these
methods can be adapted to count the Hamilton cycles in nanotubes of greater
(but still fixed) width, with the caveat that the resulting expressions involve
matrix powers. We also consider cyclically $4$-edge-connected cubic planar
graphs with few Hamilton cycles, and exhibit an infinite family of such graphs
each with exactly $4$ Hamilton cycles. Finally we consider the "other extreme"
for these two classes of graphs, thus investigating cyclically $4$-edge
connected cubic planar graphs with many Hamilton cycles and the cyclically
$5$-edge connected cubic planar graphs with few Hamilton cycles. In each of
these cases, we present partial results, examples and conjectures regarding the
graphs with few or many Hamilton cycles.
