enumeration of the number of spanning trees in some ... - Toubkal

7.3. Formulae for the Number of Spanning Trees in a Maximal Planar Map 125n − 1 −1 −1 −1 −1 −1 . . . −1 −1−1 n − 1 −1 −1 −1 −1 . . . −1 −1−1 −1 3 −1 0 0 . . . 0 0−1 −1 −1 4 −1 0 . . . 0 0τ(E n ) =−1 −1 0 −1 4 −1 . . . 0 0−1 −1 0 0 −1 4 . . . 0 0. . . . . . . .. . .−1 −1 0 0 0 0 . . . 4 −1∣ −1 −1 0 0 0 0 . . . −1 4 ∣we denote r i by the i-th row and c i by the i-th column of the determinant. In previousdeterminant, we replace c 1 by c 1 + c 2 + ... + c n−1 , i.e., we add to the first column the sumof other (transformation is symbolized as follows: c 1 ← ∑ n−1i=1 c i, ; this does not changethe determinant, then we obtain:1 −1 −1 −1 −1 −1 . . . −1 −11 n − 1 −1 −1 −1 −1 . . . −1 −10 −1 3 −1 0 0 . . . 0 00 −1 −1 4 −1 0 . . . 0 0τ(E n ) =0 −1 0 −1 4 −1 . . . 0 00 −1 0 0 −1 4 . . . 0 0. . . . . ... . . .0 −1 0 0 0 0 . . . 4 −1∣1 −1 0 0 0 0 . . . −1 4 ∣Next, we replace c j by c 1 + c j for j = 2, ..., n − 1, i.e., c j ← c 1 + c j , we obtain:1 0 0 0 0 0 . . . 0 01 n 0 0 0 0 . . . 0 00 −1 3 −1 0 0 . . . 0 00 −1 −1 4 −1 0 . . . 0 0τ(E n ) =0 −1 0 −1 4 −1 . . . 0 00 −1 0 0 −1 4 . . . 0 0. . . . . . . .. . .0 −1 0 0 0 0 . . . 4 −1∣1 0 1 1 1 1 . . . 0 5 ∣Expanding L n−1 along the first row we obtain the determinant of order (n − 2) × (n − 2)