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

Counting independent sets in cubic graphs of given girth