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

76CHAPTER 4.NEW METHODS TO COMPUTE THE NUMBER OF SPANNINGTREES OF PLANAR MAPS4.3.3 Counting the number of spanning trees in a map of typeC= C 1 : C 2Now, we are interested in the maps of type C= C 1 : C 2 , C is a map such that if a pathconnects a vertex v k of C 1 and a vertex v l of C 2 must pass through v 1 or v 2 ; C 1 and C 2have two common vertices v 1 , v 2 (see Figure 4.10).Figure 4.10: A map C= C 1 : C 2Example 4.3.23 Here is an example of this type of maps (see Figure 4.11):Figure 4.11: An example of map C= C 1 : C 2Property 4.3.24 Let C be a map of type C= C 1 : C 2- C 1 and C 2 have two common vertices v 1 , v 2 and a common face (the external face).- V C =V C1 +V C2 -2, E C =E C1 +E C2 and F C =F C1 +F C2 .- A path that connects a vertex of C 1 and a vertex of C 2 must pass through v 1 or v 2 .- If we remove the two vertices v 1 and v 2 of the map C, the resulting map is not connected.Example 4.3.25 Here is an example of maps C= C 1 : C 2 , C 1 , C 2 , C 1 .v 1 v 2 and C 2 .v 1 v 2 (seeFigure. 4.12).