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

42 CHAPTER 2. SPANNING TREESFigure 2.1: A graph(left) and all spanning trees (right) of this graph.The number of spanning trees of a finite graph or multigraph G, also known as thecomplexity τ(G), is certainly one of the most important graph-theoretical parameters,and also one of the oldest. Its applications range from the theory of networks, wherethe number of spanning trees is used as a measure for network reliability [34] totheoretical chemistry, in connection with the enumeration of certain chemical isomers[22]. Kirchhof’s celebrated matrix tree theorem [65] that relates the properties of anelectrical network to the number of spanning trees in the underlying graph.The number of spanning trees in a graph (network) is an important, well studied quantity[36]. As well as being of combinatorial interest, several application uses, mentionedin the following, are adduced in [62].1. Kirchhoff’s laws, well know as Matrix Tree Theorem, provide an effective methodfor designing electrical circuits, which are enormously useful in the analysis andsynthesis of networks.2. Suppose we are given a network of communication lines, which can break. Theprobability of a single line breaking is 1−p. It is necessary to estimate the reliabilityof such a network. If the reliability is the probability of connectedness of the network,P , thenm∑P = A k p k (1 − p) m−k ,k=n−1where n is the number of vertices, m the number of edges, of the graph; A k is thenumber of connected subgraphs with n vertices and k edges. It is clear that, if thereliability of each line is small, thenP ≈ A n−1 p n−1 (1 − p) m−n+1 ,where A n−1 is the number of spanning trees of the graph.