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

4.3. Main Results 69one vertex (any path connecting a vertex of C 1 to a vertex of C 2 must pass through thisvertex); see Figure 4.3.Figure 4.3: A map C = C 1 • C 2Property 4.3.2 Let C be a map of type C = C 1 • C 2- C 1 and C 2 have a common vertex v 1 and a common face (the external face).- V C =V C1 +V C2 -1, E C =E C1 +E C2 and F C =F C1 +F C2 -1- A path from a vertex of C 1 to a vertex of C 2 must pass through v 1 .- If we remove the vertex v 1 of the map C, the resulting map is not connected.Theorem 4.3.3 If we have a map C such that C = C 1 • C 2 , thenτ(C) = τ(C 1 • C 2 ) = τ(C 1 ) × τ(C 2 ).Proof: Each path that connects a vertex of C 1 to a vertex of C 2 must pass through v 1 .The laplacian matrix associated with a previous map C = C 1 • C 2 is as follows:After deleting the row and the column of the vertex v 1 , we obtain the following matrix:⎛⎞M n1 ,n 10⎝⎠0 M n2 ,n 2