Counting independent sets in cubic graphs of given girth

TL;DR

This paper establishes tight bounds on the number of independent sets in cubic graphs with specified girth, identifying extremal graphs like the Heawood and Petersen graphs, and proposing a broader conjecture about Moore graphs.

Contribution

It provides the first tight bounds on independent sets in cubic graphs of given girth and introduces a conjecture linking Moore graphs to extremal independent set counts.

Findings

Tight upper bound on independent sets for cubic graphs with girth ≥ 5.

Tight lower bound on independent sets for triangle-free cubic graphs.

Heawood and Petersen graphs are extremal examples.

Abstract

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.