  We prove a tight upper bound on the independence polynomial (and total number
of independent sets) of cubic graphs of girth at least 5. The bound is achieved
by unions of the Heawood graph, the point/line incidence graph of the Fano
plane.
  We also give a tight lower bound on the total number of independent sets of
triangle-free cubic graphs. This bound is achieved by unions of the Petersen
graph.
  We conjecture that in fact all Moore graphs are extremal for the scaled
number of independent sets in regular graphs of a given minimum girth,
maximizing this quantity if their girth is even and minimizing if odd. The
Heawood and Petersen graphs are instances of this conjecture, along with
complete graphs, complete bipartite graphs, and cycles.
