▁of ▁another ▁simple ▁transformation : ▁ ▁T ut te ’ s ▁original ▁definition ▁of ▁ ▁is ▁equivalent ▁but ▁less ▁easily ▁stated . ▁For ▁connected ▁ ▁we ▁set ▁ ▁where ▁ ▁denotes ▁the ▁number ▁of ▁sp anning ▁trees ▁of ▁internal ▁activity ▁and ▁external ▁activity ▁. ▁ ▁A ▁third ▁definition ▁uses ▁a ▁delet ion – cont raction ▁rec urrence . ▁The ▁edge ▁contra ction ▁ ▁of ▁graph ▁ ▁is ▁the ▁graph ▁obtained ▁by ▁mer ging ▁the ▁vertices ▁ ▁and ▁ ▁and ▁removing ▁the ▁edge ▁. ▁We ▁write ▁ ▁for ▁the ▁graph ▁where ▁the ▁edge ▁ ▁is ▁merely ▁removed . ▁Then ▁the ▁T ut te ▁polynomial ▁is ▁defined ▁by ▁the ▁rec urrence ▁relation ▁ ▁if ▁ ▁is ▁neither ▁a ▁loop ▁nor ▁a ▁bridge , ▁with ▁base ▁case ▁ ▁if ▁ ▁contains ▁ ▁brid ges ▁and ▁ ▁loops ▁and ▁no ▁other ▁edges . ▁Es pecially , ▁ ▁if ▁ ▁contains ▁no ▁edges . ▁ ▁The ▁random ▁cluster ▁model ▁from ▁statistical ▁mechan ics ▁due ▁to ▁ ▁provides ▁yet ▁another ▁equivalent ▁definition . ▁The ▁polynomial ▁ ▁is ▁equivalent ▁to ▁ ▁under ▁the ▁transformation ▁ ▁Properties ▁The ▁T ut te ▁polynomial ▁factors ▁into ▁connected ▁components . ▁If ▁ ▁is ▁the ▁union ▁of ▁dis joint ▁graphs ▁ ▁and ▁ ▁then ▁▁▁ ▁If ▁ ▁is ▁plan ar ▁and ▁ ▁denotes ▁its ▁dual ▁graph ▁then ▁▁▁▁ ▁Es pecially , ▁the ▁chrom atic ▁polynomial ▁of ▁a ▁plan ar ▁graph ▁is ▁the ▁flow ▁polynomial ▁of ▁its ▁dual . ▁T ut te ▁refers ▁to ▁such ▁functions ▁as ▁V - functions . ▁ ▁Ex amples ▁Is omorphic