▁Pot ts ▁model ▁from ▁statistical ▁physics . ▁It ▁is ▁also ▁the ▁source ▁of ▁several ▁central ▁computational ▁problems ▁in ▁theoretical ▁computer ▁science . ▁ ▁The ▁T ut te ▁polynomial ▁has ▁several ▁equivalent ▁definitions . ▁It ▁is ▁equivalent ▁to ▁Whit ney ’ s ▁rank ▁polynomial , ▁T ut te ’ s ▁own ▁dich rom atic ▁polynomial ▁and ▁Fort u in – K aste le yn ’ s ▁random ▁cluster ▁model ▁under ▁simple ▁transformations . ▁It ▁is ▁essentially ▁a ▁generating ▁function ▁for ▁the ▁number ▁of ▁edge ▁sets ▁of ▁a ▁given ▁size ▁and ▁connected ▁components , ▁with ▁immediate ▁general izations ▁to ▁mat roid s . ▁It ▁is ▁also ▁the ▁most ▁general ▁graph ▁invariant ▁that ▁can ▁be ▁defined ▁by ▁a ▁delet ion – cont raction ▁rec urrence . ▁Several ▁text books ▁about ▁graph ▁theory ▁and ▁mat roid ▁theory ▁dev ote ▁entire ▁chap ters ▁to ▁it . ▁ ▁De fin itions ▁Definition . ▁For ▁an ▁und irect ed ▁graph ▁ ▁one ▁may ▁define ▁the ▁T ut te ▁polynomial ▁as ▁ ▁where ▁ ▁denotes ▁the ▁number ▁of ▁connected ▁components ▁of ▁the ▁graph ▁. ▁In ▁this ▁definition ▁it ▁is ▁clear ▁that ▁ ▁is ▁well - defined ▁and ▁a ▁polynomial ▁in ▁ ▁and ▁. ▁ ▁The ▁same ▁definition ▁can ▁be ▁given ▁using ▁slightly ▁different ▁notation ▁by ▁letting ▁ ▁denote ▁the ▁rank ▁of ▁the ▁graph ▁. ▁Then ▁the ▁Whit ney ▁rank ▁generating ▁function ▁is ▁defined ▁as ▁ ▁The ▁two ▁functions ▁are ▁equivalent ▁under ▁a ▁simple ▁change ▁of ▁variables : ▁ ▁T ut te ’ s ▁dich rom atic ▁polynomial ▁ ▁is ▁the ▁result