By Herbert S. Wilf
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Extra resources for Algorithms and Complexity (Second edition)
Most counting problems on graphs are much easier for labeled than for unlabeled graphs. Consider the following question: How many graphs are there that have exactly n vertices? Suppose first¡ that we mean labeled graphs. A graph of n vertices has ¢ a maximum of n2 edges. To construct a graph, we would decide which ¡ ¢ of these possible edges would be used. We can make each of these n2 decisions independently, and for every way of deciding where to put the edges, we would get a diﬀerent graph. Therefore, the number of labeled n graphs of n vertices is 2( 2 ) = 2n(n−1)/2 .
It follows that: yn = y0 + n X dj j=1 (n ≥ 0). 29) to reverse the change of variables to get back to the original unknowns x0 , x1 , . , and find that: n o n X dj xn = (b1 b2 · · · bn ) x0 + j=1 (n ≥ 1). 31) It is not recommended that the reader memorize the solution that we have just obtained. It is recommended that the method by which the solution was found be mastered. 29), then (b) solve that one by summation and (c) go back to the originalunknowns. As an example, consider the first-order equation: (n ≥ 0; x0 = 0).
If a graph G is given, then thanks to the following elegant theorem of Euler, it is quite easy to decide whether or not G has an Eulerian path. In fact, the theorem applies also to multigraphs, which are graphs except that they are allowed to have several diﬀerent edges joining the same pair of vertices. 12. A (multi-)graph has an Eulerian circuit (resp. path) if and only if it is connected and has no (resp. has exactly two) vertices of odd degree. 4. Not every graph has a Hamiltonian path. 4(b) doesn’t.
Algorithms and Complexity (Second edition) by Herbert S. Wilf