# Radio Number for Friendship Communication Networks
## 2 Radio Number of Friendship Graph  
In this section, we seek to find the upper bound of \(rn(G)\) where \(G\) is a friendship graph. **Definition 1**. : _For the given positive integers \(k,\ m,\) a friendship graph, denoted as \(F_{m,k}\), is represented as \(k\) cycles (blocks), each of length \(m\), and all have one common vertex. For an illustration, \(F_{3,k}\) is shown in Figure 3._  
**Definition 2**. : _The order of the graph \(G\) is the cardinality of its vertex set \(V(G)\)._  
**Theorem 1**. : _The radio number of the friendship graph \(F_{3,k}\) is its order._  
Proof. : Following Definition 1 for \(F_{3,k}\), we find that \(diam(F_{3,k})=2\), \(\left|V(F_{3,k})\right|=2k+1\), and \(d\left(x_{i},\ x_{j}\right)\geq 1\) for any \(x_{i},\ x_{j}\in V(F_{3,k})\) and \(i\
eq j\). Since any radio labeling \(L\) is one-to-one, it follows that  
\[rn(F_{3,k})\geq\left|V(F_{3,k})\right| \tag{1}\]  
Figure 3: \(F_{3,k}\) with labeling of vertices. Figure 2: Friendship graph for modeling frequency interference of stations A, B, C, D, E, F and G. (**a**) Many transmitting stations and more Physical frequency interference. (**b**) Friendship interference graph. Define the map \(L\) with codomain \(\{0,2,3,\cdots,2k+1\}\) as follows:  
\[ \begin{array}{c}L(x_{1})=0;\\ L(x_{i})=i;2\leq i\leq 2k+1\end{array} \]  
Now, we claim to prove that \(\big{|}L(x_{i})-L\big{(}x_{j}\big{)}\big{|}\geq diam(F_{3,k})+1-d\big{(}x_{i},x _{j}\big{)},\) that is  
\[\big{|}L(x_{i})-L\big{(}x_{j}\big{)}\big{|}\geq 3-d\big{(}x_{i},\ \ x_{j}\big{)} \text{ for all }x_{i},\ x_{j}\in V(F_{3,k})\ \text{ and }i\
eq j.\]  
**Case 1. ** Let \(2\leq i\leq k+1\) and \(2\leq j\leq k+1.\) Then, \(\big{|}L(x_{i})-L\big{(}x_{j}\big{)}\big{|}=|i-j|\geq 1.\)  
Since \(d\big{(}x_{i},x_{j}\big{)}=2,\) then, \(\big{|}L(x_{i})-L\big{(}x_{j}\big{)}\big{|}\geq 3-d\big{(}x_{i},x_{j}\big{)}.\)  
**Case 2. ** Let \(i=1,j\in\{2,3,\cdots,2k+1\}.\) Then, \(\big{|}L(x_{i})-L\big{(}x_{j}\big{)}\big{|}=|0-j|=j\geq 2.\)  
Since \(d\big{(}x_{i},x_{j}\big{)}=1.\) Consequently, \(\big{|}L(x_{i})-L\big{(}x_{j}\big{)}\big{|}\geq 3-d\big{(}x_{i},x_{j}\big{)}.\)  
**Case 3. ** Let \(i,j\in\{k+2,k+3,\cdots,2k+1\}.\) Then, \(\big{|}L(x_{i})-L\big{(}x_{j}\big{)}\big{|}\geq 1.\) Since, \(d\big{(}x_{i},x_{j}\big{)}=2.\) Therefore, \(\big{|}L(x_{i})-L\big{(}x_{j}\big{)}\big{|}\geq 3-d\big{(}x_{i},x_{j}\big{)}.\)  
**Case 4. ** Let \(i\in\{2,3,\cdots,k+1\},\)\(j=k+i\). Then, \(\big{|}L(x_{i})-L\big{(}x_{j}\big{)}\big{|}=|k+i-i|=k\geq 2.\) Since \(d\big{(}x_{i},x_{j}\big{)}=1\) then \(\big{|}L(x_{i})-L\big{(}x_{j}\big{)}\big{|}\geq 3-d\big{(}x_{i},x_{j}\big{)}.\)  
**Case 5. ** Let \(i\in\{2,3,\cdots,k+1\},\)\(j\in\{k+2,k+3,\cdots,2k+1\},\) for every \(j\
eq k+i,\)  
\(\big{|}L(x_{i})-L\big{(}x_{j}\big{)}\big{|}=|i-j|\geq k-1\) where \(k\geq 2.\) Moreover, \(d\big{(}x_{i},x_{j}\big{)}=2.\)  
Hence, \(\big{|}L(x_{i})-L\big{(}x_{j}\big{)}\big{|}\geq 3-d\big{(}x_{i},x_{j}\big{)}.\)  
Thus, \(L\) is a radio labeling of \(F_{3,k}\) and  
\[rn(F_{3,k})\leq 2k+1 \tag{2}\]  
From Formulas (1) and (2), \(rn(F_{3,k})=2k+1.\)\(\square\)  
For more illustrations, Figure 3 shows \(F_{3,\ k}\) with labeling of vertices. **Theorem 2. **_Let \(k>2\) and \(G\cong F_{4,\ k}\) be a friendship graph with blocks each of length 4 and \(\big{|}V(F_{4,k}\big{)}\big{|}=3k+1\) then \(rn(F_{4,\ k})\leq 7k+1.\)_  
**Proof**. Define the map \(L\) as follows:  
\[L\Big{(}x_{jk+i}\Big{)}=\left\{ \begin{array}{cc}0,&i=j=0\\ 3+i-1,&j=0,1\leq i\leq k\\ k+1+3i,&j=1,1\leq i\leq k\\ 4k+1+3i,&j=2,1\leq i\leq k\end{array} \right.\]  
Since \(diam(F_{4,k})=4,\) we claim to prove that \(\big{|}L(x_{u})-L(x_{v})\big{|}\geq 5-d(x_{u},x_{v})\) for all \(x_{u},x_{v}\in V(F_{4,k})\) and \(u\
eq v.\)  
**Case 1. ** Let \(j=0,1\leq i\leq k,\) then \(\big{|}L(x_{0})-L\Big{(}x_{jk+i}\Big{)}\big{|}=\big{|}0-(3+i-1)\big{|}=2+i.\) Since \(d\Big{(}x_{0},x_{jk+i}\Big{)}=2.\) Consequently, \(\big{|}L(x_{0})-L\Big{(}x_{jk+i}\Big{)}\big{|}\geq 5-d\Big{(}x_{0},x_{jk+i} \Big{)}.\)  
**Case 2.