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

ConclusionThis thesis presents our contribution in the research area of calculating the number ofspanning trees in connected planar maps (graphs embedded into surfaces without edgecrossings).In this thesis, we have surveyed the several methods as the Matrix Tree Theorm, thedeletion - contraction theorem, Cayley’s Theorem and other methods that can calculatethe number of spanning trees in planar maps. However, we noticed that, the calculation ofthe number of spanning trees of planar maps by using the determinant of Laplacian matrixor the Laplace spectrum is tedious and impractical, therefore, we gave new methods inthe Chapter 4 which facilitate the calculation of the number of spanning trees in planarmaps. Then, we applied these methods on certain planar maps and as a result, weobtained explicit formulae to find the number of spanning trees exactly in some specialplanar maps.The first three chapters are devoted on definitions and general properties of basicobjects studied (graphs, maps, trees, spanning trees, complexity, matrices associated to agraph, etc), and some methods for counting spanning trees are introduced.In Chapter 4, we gave some new methods to calculate the number of spanning treesin planar maps in general then in particular in some special planar maps.The fifth Chapter is devoted on derivative explicit formulae which facilitate the calculationof the number of spanning trees in some families of planar maps which are called then-Fan chains, the n- Grid chains, the n-tent chains, the n-Hexagonal chains, the n-Eightchains, the n-Home chains, the n-Barrel chains, the n-Light chains, the n-Kite chains,the n-Envelope chains and the n-Diphenylene, ... etc. These results obtained came as aresult of the use of our methods that we have derived in Chapter 4.In chapter 6, we focused on a particular type of planar maps which is called the starflower planar map, and we derived an explicit formula which calculates the number ofspanning trees in this type of planar maps.Finally, in Chapter 7, we were interested in the maximal planar map. We calculatedthe Wiener index in general in the case of planar maps, in particular, in the maximalplanar maps and gave an inequality, which minimizes and maximizes any planar map bythe maximal planar map with n vertices and the path with n vertices. We then derivedan explicit formula to calculate the number of spanning trees in the maximal planar map,then we have deduced the formula for the number of spanning trees in the crystal planarmap.133