ut te ▁polynomial . ▁ ▁Special is ations ▁At ▁various ▁points ▁and ▁lines ▁of ▁the ▁- plane , ▁the ▁T ut te ▁polynomial ▁evalu ates ▁to ▁quantities ▁that ▁have ▁been ▁studied ▁in ▁their ▁own ▁right ▁in ▁diverse ▁fields ▁of ▁mathematics ▁and ▁physics . ▁Part ▁of ▁the ▁appeal ▁of ▁the ▁T ut te ▁polynomial ▁comes ▁from ▁the ▁un ifying ▁framework ▁it ▁provides ▁for ▁anal ys ing ▁these ▁quantities . ▁ ▁Ch rom atic ▁polynomial ▁ ▁At ▁, ▁the ▁T ut te ▁polynomial ▁special ises ▁to ▁the ▁chrom atic ▁polynomial , ▁ ▁where ▁ ▁denotes ▁the ▁number ▁of ▁connected ▁components ▁of ▁G . ▁ ▁For ▁integer ▁ λ ▁the ▁value ▁of ▁chrom atic ▁polynomial ▁ ▁equals ▁the ▁number ▁of ▁vertex ▁colour ings ▁of ▁G ▁using ▁a ▁set ▁of ▁ λ ▁colours . ▁It ▁is ▁clear ▁that ▁ ▁does ▁not ▁depend ▁on ▁the ▁set ▁of ▁colours . ▁What ▁is ▁less ▁clear ▁is ▁that ▁it ▁is ▁the ▁evaluation ▁at ▁ λ ▁of ▁a ▁polynomial ▁with ▁integer ▁coefficients . ▁ ▁To ▁see ▁this , ▁we ▁observe : ▁ ▁If ▁G ▁has ▁n ▁vertices ▁and ▁no ▁edges , ▁then ▁. ▁ ▁If ▁G ▁contains ▁a ▁loop ▁( a ▁single ▁edge ▁connecting ▁a ▁vertex ▁to ▁itself ), ▁then ▁. ▁ ▁If ▁e ▁is ▁an ▁edge ▁which ▁is ▁not ▁a ▁loop , ▁then ▁ ▁The ▁three ▁conditions ▁above ▁enable ▁us ▁to ▁calculate ▁, ▁by ▁applying ▁a ▁sequence ▁of ▁edge ▁delet ions ▁and ▁contra ctions ; ▁but ▁they ▁give ▁no ▁guarantee ▁that ▁a ▁different ▁sequence ▁of ▁delet ions ▁and ▁contra ctions ▁will