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

ENUMERATION OF THE NUMBER OF SPANNING TREESIN SOME SPECIAL PLANAR MAPSAbstractThe number of spanning trees in a planar map - graph embedded into the surfaceswithout edge-crossings - (network) is an important well-studied quantity and invariantof the graph (network); moreover it is also an important measure of reliability of anetwork which plays a central role in Kirchhoff’s classical theory of electrical networks.In a graph (network), that contains several cycles, we must remove the redundanciesin this network, i.e., we obtain a spanning tree. A spanning tree in a map C is a treewhich has the same vertex set as C (tree that passing through all the vertices of the map C).Our research theme in this thesis focuses on the counting of the number of spanningtrees in connected planar maps, a subject in combinatorial graph theory; as well as, tofind new methods to calculate the number of spanning trees in any map (network).Spanning trees are relevant to various aspects of graphs. Generally, the number ofspanning trees in a network can be obtained by computing a related determinant ofthe Laplacian matrix or computing the Laplace spectrum of the network. However,for a large map, evaluating the relevant determinant is computationally intractable.In this work, we provide new methods to facilitate the calculation of the number ofspanning trees in planar maps and to prove new simplified and generalized results.Finally, we apply these methods on certain planar maps to derive several explicit formulaefor calculating the number of spanning trees in some special families of planar maps.Keywords: graphs, maps, trees, spanning trees, complexity, Laplacian matrix,Matrix-Tree Theorem, the n-Fan chains, the n- Grid chains, star flower planar map,maximal planar map, Wiener index.