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

2.2. Basic concepts and research background 43Thus, with low reliability of each of the line, the network’s reliability is determined,basically, by the number of spanning trees of the network.3. In building a maser, one must investigate the possible particle transitions. For this,one constructs a graph in which the vertices correspond to energy levels and edges topossible particle transitions. Then for the analysis of the maser’s energetics, it turnsout to be very useful to know the number of spanning trees in the correspondinggraph.The research of the number of spanning trees in a graph has a long history. Thecornerstone of the research, the Matrix Tree Theorem, dated back to 1847, is attributedto Kirchhoff. Most research about the number of spanning trees is devoted to determiningexact formulae for the number of spanning trees in many kinds of special graphs, see[6, 14, 62, 63, 65, 66, 127]. We now state the general methods for counting the numberof spanning trees in graphs in the following.Now, the problem is: given a map (graph embedded into surfaces), how manyspanning trees does it have? As example, a graph G with all its spanning trees isdisplayed in Figure 2.2.Figure 2.2: A graph G gives rise to five spanning treesLet G be a connected graph. Then a spanning tree in G is a subgraph of G thatincludes every vertex and is also a tree. For example, the following diagram shows agraph G and three of its spanning trees; (see Figure. 2.3).Figure 2.3: A graph G and three of its spanning trees