Question

Find the number of spanning trees of K3,3 using Kirchoff's matrix-tree theorem.

Answer 1

K33 is de Now the Adjaceney matrix Adjaceney matrix of the graph KB3 is. a, a b c d e f blooo 11 0 0 1 1 А | op el 0 al - - - Ooo 0 1 1 1 1 eli 1 0 0 0 0 0 Now applying Kirchoff's theorem applying we get the new matrix сь ее у al3 0 0 1 - 1 - 1 o 3 0 - - - - ع Lai -1 له ه !!! -1 3 0 0 <?o 30 0 0 31 Now the cofactor of (a, a). i lo factor for all 7 I elements will be some by kirchoff's law 3 0 - -1 - 0 3 - -1 -1 [ 7 moo -1 -1-1 0 0 3 0 o 31 0 -3 -1 -1 C,+ 385 @z-C5 Cu-C5 I 0 0 0 3 o o -1 - 3 o o -1 3 0 - -3 -3 3. - 3 3 o o -1 -1 3 0 -1 -1 0 3 8 -1 -3 -3 - 3 0 0 0 -1 -2 3 o wwoo wwoo = (-1) [cate, 1 8 7 -3

1 -2 NN 3 o = (-1)(-3) . омм wow T7 -3 -3 - 3 - 2 3 w 0 1 T 15 -3 7 -3 o -3 | Rg + R3 = 3(-3) { (-2)*3) - 5x3} = (-9) 46-15) = (-9) (9) - 81 Hence the number of spanning trees of K33 is 81.