**8. Conclusions**

In the present chapter, some problems of combinatorial graph enumeration as well as some useful techniques for obtaining a closed mathematical expression were addressed. When it is not possible to obtain a closed expression, asymptotic estimations of the kind used in analytic combinatorics can be used. In Section 10 the use of these techniques for proving a result in the field of virus spreading control was illustrated [31–34].

This allowed to explore the application of combinatorial techniques to control problems in networks and thus verify the goodness of said methods for network analysis and the control of virus propagation in them. This still needs to be studied by applying the combinatorial methods discussed above, if there are any other types of topologies that prevent the application of the selection criteria of nodes to be controlled under the hypothesis of behavior of partially heterogeneous nodes, that is, if in the network we have subsets of nodes with the same behavioral parameters. The selecting node criteria described in [30] are based on the combination of the parameter values of the selected nodes as well as their degrees. In many recent publications [35–42], interesting and elaborated methods for detecting the influencer nodes in complex networks have been proposed that I will try to apply in combination with the mentioned criteria in the future in order to reduce the number of nodes to be controlled.

**Author details**

**241**

Carlos Rodríguez Lucatero

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

Departamento de Tecnologías de la Información, Universidad Autónoma

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Metropolitana Unidad Cuajimalpa, Mexico City, Mexico

provided the original work is properly cited.

\*Address all correspondence to: profesor.lucatero@gmail.com

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

and, replacing *c*<sup>1</sup> by *r* in (59), we get

*Probability, Combinatorics and Control*

controlled are almost always applicable.

**8. Conclusions**

was illustrated [31–34].

ber of nodes to be controlled.

**240**

tioned criteria are almost always applicable.

lim*n*!∞

Therefore, if the degree *<sup>r</sup>* is constant, the lim*n*!<sup>∞</sup> *Lr*

ffiffi 2 p

Now, our main result can be stated as a consequence of Theorem 1.11.

nodes have homogeneous behavior, then the criteria for selecting nodes to be

*rrne <sup>r</sup>*ð Þ <sup>2</sup>þ4*rn*þ<sup>1</sup> *<sup>=</sup>*<sup>4</sup>

2 *n n*ð Þ �1 2

Theorem 1.12 If we assume that all graphs are uniformly distributed and that the

**Proof of Theorem 1.11.** As a consequence of Theorem 11, we know that the probability that a regular graph appears tends to zero as *n* ! ∞. Then, the men-

In the present chapter, some problems of combinatorial graph enumeration as well as some useful techniques for obtaining a closed mathematical expression were addressed. When it is not possible to obtain a closed expression, asymptotic estimations of the kind used in analytic combinatorics can be used. In Section 10 the use of these techniques for proving a result in the field of virus spreading control

This allowed to explore the application of combinatorial techniques to control problems in networks and thus verify the goodness of said methods for network analysis and the control of virus propagation in them. This still needs to be studied by applying the combinatorial methods discussed above, if there are any other types of topologies that prevent the application of the selection criteria of nodes to be controlled under the hypothesis of behavior of partially heterogeneous nodes, that is, if in the network we have subsets of nodes with the same behavioral parameters. The selecting node criteria described in [30] are based on the combination of the parameter values of the selected nodes as well as their degrees. In many recent publications [35–42], interesting and elaborated methods for detecting the

influencer nodes in complex networks have been proposed that I will try to apply in combination with the mentioned criteria in the future in order to reduce the num-

*rn=*2 *e* � �*rn=*<sup>2</sup>

*Gn* ¼ 0.

*:* (60)

ffiffiffiffiffi <sup>2</sup>*π<sup>r</sup>* <sup>p</sup> ð Þ*<sup>n</sup>*
