▁ ▁If ▁i ▁= ▁j ▁then ▁ ▁For ▁an ▁und irect ed ▁graph ▁ ▁General ▁sum ▁rule ▁For ▁any ▁N - ver tex ▁simple ▁connected ▁graph ▁G   =   ( V ,   E ) ▁and ▁arbitrary ▁N × N ▁matrix ▁M : ▁ ▁From ▁this ▁generalized ▁sum ▁rule ▁a ▁number ▁of ▁relationships ▁can ▁be ▁derived ▁depending ▁on ▁the ▁choice ▁of ▁M . ▁Two ▁of ▁note ▁are ; ▁ ▁where ▁the ▁ ▁are ▁the ▁non - zero ▁eigenvalues ▁of ▁the ▁Lap la cian ▁matrix . ▁This ▁un ordered ▁sum ▁ ▁is ▁called ▁the ▁Kirch hoff ▁index ▁of ▁the ▁graph . ▁ ▁Rel ations hip ▁to ▁the ▁number ▁of ▁sp anning ▁trees ▁of ▁a ▁graph ▁ ▁For ▁a ▁simple ▁connected ▁graph ▁G   =   ( V ,   E ), ▁the ▁resistance ▁distance ▁between ▁two ▁vertices ▁may ▁be ▁expressed ▁as ▁a ▁function ▁of ▁the ▁set ▁of ▁sp anning ▁trees , ▁T , ▁of ▁G ▁as ▁follows : ▁ ▁where ▁ ▁is ▁the ▁set ▁of ▁sp anning ▁trees ▁for ▁the ▁graph ▁. ▁ ▁As ▁a ▁squ ared ▁E uclidean ▁distance ▁Since ▁the ▁Lap la cian ▁ ▁is ▁symmetric ▁and ▁positive ▁semi - defin ite , ▁its ▁pseudo in verse ▁ ▁is ▁also ▁symmetric ▁and ▁positive ▁semi - defin ite . ▁ ▁Thus , ▁there ▁is ▁a ▁ ▁such ▁that ▁ ▁and ▁we ▁can ▁write : ▁ ▁showing ▁that ▁the ▁square ▁root ▁of ▁the ▁resistance ▁distance ▁corresponds ▁to ▁the ▁E uclidean ▁distance ▁in ▁the ▁space ▁sp anned ▁by ▁. ▁ ▁Connection ▁with ▁F ib