ts ▁model . ▁Vari ous ▁physical ▁quantities ▁anal ys ed ▁in ▁the ▁framework ▁of ▁the ▁Pot ts ▁model ▁translate ▁to ▁specific ▁parts ▁of ▁the ▁. ▁ ▁Flow ▁polynomial ▁ ▁At ▁, ▁the ▁T ut te ▁polynomial ▁special ises ▁to ▁the ▁flow ▁polynomial ▁studied ▁in ▁combin ator ics . ▁For ▁a ▁connected ▁and ▁und irect ed ▁graph ▁G ▁and ▁integer ▁k , ▁a ▁now here - zero ▁k - flow ▁is ▁an ▁assignment ▁of ▁“ flow ” ▁values ▁ ▁to ▁the ▁edges ▁of ▁an ▁arbitrary ▁orientation ▁of ▁G ▁such ▁that ▁the ▁total ▁flow ▁entering ▁and ▁leaving ▁each ▁vertex ▁is ▁con gru ent ▁mod ulo ▁k . ▁The ▁flow ▁polynomial ▁ ▁denotes ▁the ▁number ▁of ▁now here - zero ▁k - flow s . ▁This ▁value ▁is ▁int imately ▁connected ▁with ▁the ▁chrom atic ▁polynomial , ▁in ▁fact , ▁if ▁G ▁is ▁a ▁plan ar ▁graph , ▁the ▁chrom atic ▁polynomial ▁of ▁G ▁is ▁equivalent ▁to ▁the ▁flow ▁polynomial ▁of ▁its ▁dual ▁graph ▁ ▁in ▁the ▁sense ▁that ▁ ▁Theorem ▁( T ut te ). ▁ ▁The ▁connection ▁to ▁the ▁T ut te ▁polynomial ▁is ▁given ▁by : ▁ ▁R eli ability ▁polynomial ▁ ▁At ▁, ▁the ▁T ut te ▁polynomial ▁special ises ▁to ▁the ▁all - term inal ▁reli ability ▁polynomial ▁studied ▁in ▁network ▁theory . ▁For ▁a ▁connected ▁graph ▁G ▁remove ▁every ▁edge ▁with ▁probability ▁p ; ▁this ▁models ▁a ▁network ▁subject ▁to ▁random ▁edge ▁fail ures . ▁Then ▁the ▁reli ability ▁polynomial ▁is ▁a ▁function ▁, ▁a ▁polynomial ▁in ▁p , ▁that