**2. Body of the manuscript**

The chapter will have the following structure. In Section 3, we will make a quick revision of some mathematical tools that are frequently used for solving combinatorial enumeration problems. The first tool described will be the ordinary generating functions as well as the exponential generating functions. These tools are used when a closed form solution for an enumeration problem can be obtained. The second tool that will be described in this section will be the analytic combinatorics method that is normally used for those enumeration problems whose closed mathematical form are hard to be calculated. The analytic combinatorics techniques

## *Combinatorial Enumeration of Graphs DOI: http://dx.doi.org/10.5772/intechopen.88805*

functions is a tool that allows to relate discrete analysis proper to discrete mathe-

mathematical object complies with certain property is greater than zero.

algebra, computational sciences, or information theory.

be covered in the following sections of the present chapter.

The chapter will have the following structure. In Section 3, we will make a quick revision of some mathematical tools that are frequently used for solving combinatorial enumeration problems. The first tool described will be the ordinary generating functions as well as the exponential generating functions. These tools are used when a closed form solution for an enumeration problem can be obtained. The second tool that will be described in this section will be the analytic combinatorics method that is normally used for those enumeration problems whose closed mathematical form are hard to be calculated. The analytic combinatorics techniques

**2. Body of the manuscript**

**222**

One of the most beautiful and ingenious applications of combinatorial enumeration is the probabilistic method. The probabilistic method is intimately related to the important role that randomness plays in the field of theoretical computer science. The utility and beauty of the probabilistic method consists of being an indirect or nonconstructive proof method. This method has been used successfully during the last 60 years and constitutes one of the most important scientific contributions of the great Hungarian mathematician Erdös [1, 2]. Commonly, this method has been used to prove the existence of a certain mathematical object by showing that if we choose some object of a given class in a random way, the probability that this

The probabilistic method has been used with great success to obtain important results in fields as diverse as number theory, combinatorics, graph theory, linear

One aim of combinatorial analysis is to count the different ways of arranging objects under given constraints. Sometimes the structures to be counted are finite and some other times they are infinite. To enumerate is very important in many scientific fields because it allows to evaluate and compare different solutions to a given problem. For example, in computer science, if I want to compare different algorithms that solve a given kind of information processing problem, it is necessary to enumerate the number of steps taken by each one of them in the worst case and to choose the one whose performance is the best. The task of enumerating things can evolve in complexity to some point that the elementary arithmetical operations are not enough to reach the goal. For that reason many enumeration problems have inspired the most talented mathematicians for developing very ingenious methods for solving them. Some of these techniques for solving combinatorics enumeration problems are going to be exposed in the next section. One very interesting subject of the discrete mathematics is the graph theory. It is in the graph theory where many of the most interesting enumerations of graphs arise that have some given structural property that require the utilization of mathematical tools that facilitate the discovery of closed mathematical forms for the calculation of the number of graphs that accomplish some topological property. In this context it can be important to enumerate how many labeled graphs can be constructed with *n* vertex or how many connected graphs with *n* vertex exist, etc. For some of these problems, clever methods have been devised for their calculation that lead to closed mathematical forms by the use of generating functions. For other problems, it is very hard to obtain a closed mathematical form solution, and in that case some asymptotic methods have been developed for estimating a bound when the number of vertex is very large by means of the Cauchy theorem. These mathematical tools are going to

matics with continuous analysis.

*Probability, Combinatorics and Control*

allow to estimate an upper bound of that kind of enumeration problems. We take the enumeration of labeled trees as an example of combinatorial enumeration problem that can be solved in a closed mathematical form and make an overview of some different methods devised for that end in Section 4. In Section 5, we roughly describe how the generating functions can be used to solve graph enumeration problems. In Section 6, we take the problem of enumerating regular graphs as an example of a problem whose closed mathematical form is hard to obtain and apply the analytic combinatorics techniques for estimating an upper bound. In Section 7, I give an application of the combinatorial enumeration for proving the almost sure applicability of a node selecting criteria for controlling virus spreading in a complex network. Finally in Section 8, we make some comments about the possible future applications of the combinatorial enumeration methods.
