# Radio Number for Friendship Communication Networks
## 4 Computational Study  
We carried out a computational study to measure the efficiency of the proposed upper bounds by Theorems 1-4 compared to the results of the algorithms introduced in [3; 4]. Moreover, the comparison between the results of those Theorems and the mathematical model introduced in Section 3 is also presented. All of these are compatible with a PC with a Core i7 [EMAIL_ADDRESS] GHz, 8 GB of RAM, and a 64-bit operating system. The computational studies were carried out using MATLAB R2016a and the MS Windows 7 Professional system. According to the upper bound of the radio number of \(F_{3,k}\), Table 1 and Figure 7 show that the proposed results in Theorem 1 outperform the proposed results in [3] for \(k\) when it is odd. When \(k\) is even, the same results occur. On the other hand, the proposed results in Theorem 1 outperform the proposed results in [4] for every \(k\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{2}{c}{_Saha_[3]} & \multicolumn{2}{c}{_ILPM_[4]} & \multicolumn{2}{c}{_Theorem 1_} \\ \cline{3-6} \(k\) & \(n\) & \(m(F_{3,k})\) & _CPU Time_ & \(m(F_{3,k})\) & _CPU Time_ & \(m(F_{3,k})\) \\ \hline
1 & 3 & 4 & 0.024824 & 4 & 0.07009 & 3 \\ \hline
2 & 5 & 5 & 0.026428 & 7 & 0.056758 & 5 \\ \hline
3 & 7 & 8 & 0.028891 & 10 & 0.060813 & 7 \\ \hline
4 & 9 & 9 & 0.035933 & 13 & 0.066483 & 9 \\ \hline
5 & 11 & 12 & 0.035994 & 16 & 0.067994 & 11 \\ \hline
6 & 13 & 13 & 0.03913 & 19 & 0.07225 & 13 \\ \hline
7 & 15 & 16 & 0.03958 & 22 & 0.073795 & 15 \\ \hline
8 & 17 & 17 & 0.041326 & 25 & 0.078172 & 17 \\ \hline
9 & 19 & 20 & 0.044408 & 28 & 0.078172 & 19 \\ \hline
10 & 21 & 21 & 0.045015 & 31 & 0.080299 & 21 \\ \hline
11 & 23 & 24 & 0.047039 & 34 & 0.088287 & 23 \\ \hline
12 & 25 & 25 & 0.047221 & 37 & 0.102373 & 25 \\ \hline
13 & 27 & 28 & 0.048065 & 40 & 0.160287 & 27 \\ \hline
14 & 29 & 29 & 0.04859 & 43 & 0.182859 & 29 \\ \hline
15 & 31 & 32 & 0.092601 & 46 & 0.272814 & 31 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: A comparison among our results, [3], and integer programming results [4] for \(F_{3,k}\). According to the running time, Table 1 and Figure 8 show that the proposed results in Theorem 1 take the constant time complexity \(O(1)\), while the proposed results in [3] take \(O(n^{3})\). On the other hand, the proposed results in Theorem 1 take less time than the proposed results in [4]. According to the upper bound of the radio number of \(F_{4,k}\), Table 2 and Figure 9 show that the proposed results in Theorem 2 outperform the proposed results in [3] for \(k=1,\ 2\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(k\)} & \multirow{2}{*}{\(n\)} & \multicolumn{2}{c}{_Saha [3]_} & \multicolumn{2}{c}{_ILPM [4]_} & \multicolumn{2}{c}{_Theorem 2_} \\ \cline{3-8}  & & \(m(F_{4,k})\) & _CPU Time_ & \(m(F_{4,k})\) & _CPU Time_ & \(m(F_{4,k})\) \\ \hline
1 & 4 & 10 & 0.025561 &  
For \(k=3,\;4,\ldots,15\), the same results occur. On the other hand, the proposed results in Theorem 2 outperform the proposed results in [4] for every \(k\). According to the running time, Table 2 and Figure 10 show that the proposed results in Theorem 2 take the constant time complexity \(O(1)\) while the proposed results in [3] take \(O(n^{3})\). On the other hand, the proposed results in Theorem 2 take less time than the proposed results in [4]  
Figure 10: A CPU time comparison among our results, the Saha algorithm, [PERSON], [PERSON], et al., 2012 [3]; and integer programming of \(F_{4,k}\) from [PERSON], et al., 2020 [4].