▁the ▁bottom ▁of ▁the ▁evaluation ▁are ▁easy ▁to ▁compute . ▁ ▁Within ▁a ▁polynomial ▁factor , ▁the ▁running ▁time ▁t ▁of ▁this ▁algorithm ▁can ▁be ▁expressed ▁in ▁terms ▁of ▁the ▁number ▁of ▁vertices ▁n ▁and ▁the ▁number ▁of ▁edges ▁m ▁of ▁the ▁graph , ▁ ▁a ▁rec urrence ▁relation ▁that ▁scales ▁as ▁the ▁F ib on acci ▁numbers ▁with ▁solution ▁ ▁The ▁analysis ▁can ▁be ▁improved ▁to ▁within ▁a ▁polynomial ▁factor ▁of ▁the ▁number ▁ ▁of ▁sp anning ▁trees ▁of ▁the ▁input ▁graph . ▁For ▁sparse ▁graphs ▁with ▁ ▁this ▁running ▁time ▁is ▁. ▁For ▁regular ▁graphs ▁of ▁degree ▁k , ▁the ▁number ▁of ▁sp anning ▁trees ▁can ▁be ▁bounded ▁by ▁ ▁where ▁ ▁so ▁the ▁delet ion – cont raction ▁algorithm ▁runs ▁within ▁a ▁polynomial ▁factor ▁of ▁this ▁bound . ▁For ▁example : ▁ ▁In ▁practice , ▁graph ▁isomorphism ▁testing ▁is ▁used ▁to ▁avoid ▁some ▁recursive ▁calls . ▁This ▁approach ▁works ▁well ▁for ▁graphs ▁that ▁are ▁quite ▁sparse ▁and ▁exhib it ▁many ▁symmet ries ; ▁the ▁performance ▁of ▁the ▁algorithm ▁depends ▁on ▁the ▁he ur istic ▁used ▁to ▁pick ▁the ▁edge ▁e . ▁ ▁Gaussian ▁elim ination ▁In ▁some ▁restricted ▁instances , ▁the ▁T ut te ▁polynomial ▁can ▁be ▁computed ▁in ▁polynomial ▁time , ▁ultimately ▁because ▁Gaussian ▁elim ination ▁efficiently ▁comput es ▁the ▁matrix ▁operations ▁determin ant ▁and ▁Pf aff ian . ▁These ▁algorithms ▁are ▁themselves ▁important ▁results ▁from ▁algebraic ▁graph ▁theory ▁and ▁statistical ▁mechan ics . ▁▁ ▁equals ▁the ▁number ▁ ▁of ▁sp anning ▁trees ▁of ▁a ▁connected ▁graph . ▁This