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In this paper, we justify by the use of Enumerative Combinatorics, that theresults obtained in \cite{Alarcon1}, where is analysed the complex dynamics ofan epidemic model to identify the nodes that contribute the most to thepropagation process and because of that are good candidates to be controlled inthe network in order to stabilize the network to reach the extinction state, isapplicable in almost all the cases. The model analysed was proposed in\cite{Gomez1} %et al. [Phys.Rev.E 84, 036105(2011)] and results obtained in\cite{Alarcon1} implies that it is not necessary to control all nodes, but onlya minimal set of nodes if the topology of the network is not regular. Thisresult could be important in the spirit of considering policies of isolation orquarantine of those nodes to be controlled. Simulation results were presentedin \cite{Alarcon1} for large free-scale and regular networks, that corroboratethe theoretical findings. In this article we justify the applicability of thecontrollability result obtained in \cite{Alarcon1} in almost all the cases bymeans of the use of Combinatorics. {\em Mathematics Subjects Classification}:05A16,34H20,58E25 {\em Keywords}: Asymptotic Graph Enumeration Problems;Network control; virus spreading.

Use of Enumerative Combinatorics for proving the applicability of an asymptotic stability result on discrete-time SIS epidemics in complex networks