A Novel Mathematical Model for Radio Mean Square Labeling Problem

Abstract

A radio mean square labeling of a connected graph is motivated by the channel assignment problem for radio transmitters to avoid interference of signals sent by transmitters. It is an injective map $h$ from the set of vertices of the graph $G$ to the set of positive integers $\mathbb{N}$ , such that for any two distinct vertices $x,y$ , the inequality $d\left(x,y\right)+⌈{\left(h\left(x\right)\right)}^2+{\left(h\left(y\right)\right)}^2/2⌉\ge \dim \left(G\right)+1$ holds. For a particular radio mean square labeling $h$ , the maximum number of $h\left(v\right)$ taken over all vertices of $G$ is called its spam, denoted by $\mathrm{rmsn}\left(h\right)$ , and the minimum value of $\mathrm{rmsn}\left(h\right)$ taking over all radio mean square labeling $h$ of $G$ is called the radio mean square number of $G$ , denoted by $\mathrm{rmsn}\left(G\right)$ . In this study, we investigate the radio mean square numbers $\mathrm{rmsn}\left(P_n\right)$ and $\mathrm{rmsn}\left(C_n\right)$ for path and cycle, respectively. Then, we present an approximate algorithm to determine $\mathrm{rmsn}\left(G\right)$ for graph $G$ . Finally, a new mathematical model to find the upper bound of $\mathrm{rmsn}\left(G\right)$ for graph $G$ is introduced. A comparison between the proposed approximate algorithm and the proposed mathematical model is given. We also show that the computational results and their analysis prove that the proposed approximate algorithm overcomes the integer linear programming model (ILPM) according to the radio mean square number. On the other hand, the proposed ILPM outperforms the proposed approximate algorithm according to the running time.