Upper bounds for perfect matchings in pfaffian and planar graphs
  We give upper bounds on weighted perfect matchings in pfaffian graphs. These upper bounds are better than Bregman's upper bounds on the number of perfect matchings. We show that some of our upper bounds are sharp for 3 and 4-regular pfaffian graphs. We apply our results to fullerene graphs. 
Afshin Behmaram and Shmuel Friedland
math.CO math-ph math.MP