A proof of the upper matching conjecture for large graphs

We prove that the ‘Upper Matching Conjecture’ of Friedland, Krop, and Markström analogous conjecture Kahn for independent sets in regular graphs hold all large enough as a function degree. That is, every d n divisible by 2d, union n/(2d) copies complete d-regular bipartite graph maximizes number matchings size k each over on vertices. To this we utilize cluster expansion canonical ensemble statistical physics spin model, give some further applications method to maximizing minimizing given minimum girth.