▁is ▁comput able ▁in ▁polynomial ▁time ▁as ▁the ▁determin ant ▁of ▁a ▁maximal ▁principal ▁sub matrix ▁of ▁the ▁Lap la cian ▁matrix ▁of ▁G , ▁an ▁early ▁result ▁in ▁algebraic ▁graph ▁theory ▁known ▁as ▁Kirch hoff ’ s ▁Matrix – Tree ▁theorem . ▁Like wise , ▁the ▁dimension ▁of ▁the ▁b icy cle ▁space ▁at ▁ ▁can ▁be ▁computed ▁in ▁polynomial ▁time ▁by ▁Gaussian ▁elim ination . ▁ ▁For ▁plan ar ▁graphs , ▁the ▁partition ▁function ▁of ▁the ▁Is ing ▁model , ▁i . e ., ▁the ▁T ut te ▁polynomial ▁at ▁the ▁hyper bol a ▁, ▁can ▁be ▁expressed ▁as ▁a ▁Pf aff ian ▁and ▁computed ▁efficiently ▁via ▁the ▁FK T ▁algorithm . ▁This ▁idea ▁was ▁developed ▁by ▁Fish er , ▁K aste le yn , ▁and ▁Tem per ley ▁to ▁compute ▁the ▁number ▁of ▁dim er ▁covers ▁of ▁a ▁plan ar ▁lattice ▁model . ▁ ▁Mark ov ▁chain ▁Monte ▁Carlo ▁Using ▁a ▁Mark ov ▁chain ▁Monte ▁Carlo ▁method , ▁the ▁T ut te ▁polynomial ▁can ▁be ▁arbitr arily ▁well ▁approxim ated ▁along ▁the ▁positive ▁branch ▁of ▁, ▁equival ently , ▁the ▁partition ▁function ▁of ▁the ▁fer romagnet ic ▁Is ing ▁model . ▁This ▁explo its ▁the ▁close ▁connection ▁between ▁the ▁Is ing ▁model ▁and ▁the ▁problem ▁of ▁counting ▁match ings ▁in ▁a ▁graph . ▁The ▁idea ▁behind ▁this ▁celebrated ▁result ▁of ▁J err um ▁and ▁S inc la ir ▁is ▁to ▁set ▁up ▁a ▁Mark ov ▁chain ▁whose ▁states ▁are ▁the ▁match ings ▁of ▁the ▁input ▁graph . ▁The ▁trans itions