# Radio Number for Friendship Communication Networks
## 1 Introduction  
Wireless communication includes all techniques and methods of connecting and communicating between devices using a wireless signal and wireless communication technologies and gadgets. Wireless communication network services may appear in many areas, such as satellite communications, internet technology, mobile telephony, military communications, TV and radio broadcasting, and many others. Rapid development in wireless communication services led to a depletion of the most important resources and frequencies in the radio spectrum. Such development affects the economic cost of available frequencies. The reusing of frequencies may give good economies, but on the other hand, it may decrease the quality of the communication service. Using the same frequencies for many wireless communication networks leads to unacceptable interference among signals. This motivated the _frequency assignment problem_ (FAP). Given a set of transmitters in a network, the main procedure of FAP is the assignment of frequencies to transmitters, keeping interference at an acceptable level, and making use of the available frequencies in an efficient way. Such constraints of interference are related to the use of the same (or almost the same) frequencies for transmitters within a certain range from each other. The smaller the distance is among transmitters, the stronger the interference is that occurs. Therefore, it is suggested that the difference in frequency assignments should be greater. The graph theory introduces an effective model for this problem. The interference between transmitters is modeled as a graph, and this graph is called an _interference graph_\(G(V,E)\). Every vertex from \(V(G)\) stands for a unique transmitter. Any two vertices are adjacent (connected by an edge) if and only if the broadcasting of their corresponding transmitters may interfere. The frequency channels are labeled by positive integers. Hence,the vertex coloring (labeling) problem of the graph \(G\) with some constraints on the labeling is equivalent to FAP [1], where it is shown that the propagation of the signal may lead to interference in regions with a large distance from each other. As a result, not only must nearby transmitters be assigned different frequencies but they should be effectively separated. This results in the modeling of FAP as distance-constrained labeling of the graph \(G\). For some services, it is adequate that the transmitters should have distinct frequencies; moreover, the nearby transmitters inquired to use channels with appropriate separation. In this situation, FAP is equivalent to the _radio labeling problem_ of the graph \(G\) (see ref. [2]). The radio labeling problem of graph \(G\) is described as follows. Let \(G=(V(G),E(G))\) be a connected graph. For any \(u,v\in V(G)\), let \(d(u,v)\) denote the distance between two vertices \(u,v\). That is \(d(u,v)\) stands for the length of the shortest path between \(u,v\). The maximum distance between any two vertices in \(G\) is defined as the diameter of \(G\) and denoted as \(diam(G)\). Thus, \(diam(G)=max\{d(u,v):u,v\in V(G)\) }. A radio labeling of \(G\) is a one-to-one mapping \(L\) from \(V(G)\) to \(N\), where \(N\) is the set of natural number, satisfying the condition  
\[|L(u)-L(v)|\geq diam(G)+1-d(u,v).\text{ {For all }}u,\text{ }v\in V(G).\]  
The span of a labeling \(L\) is the maximum integer (span) that \(L\) assigns to a vertex in \(G\). The main objective of the radio labeling problem is to find the minimum span over all such labeling \(L\) of the graph \(G\). Such minimum span is denoted as \(rn(G)\) or the radio number of \(G\). [PERSON] [3] introduced an algorithm that determines the lower and upper bounds of the radio number of a graph. [PERSON] and [PERSON] [4] proposed the development of [PERSON]'s algorithm and introduced a novel mathematical model for the radio labeling application. The radio labeling problem has been studied for different families of graphs [5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18].