A Novel Mathematical Model for Radio Mean Square Labeling Problem

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 N , such that for any two distinct vertices x , y , the inequality d x , y + h x 2 + h y 2 / 2 ≥ dim G + 1 holds. For a particular radio mean square labeling h , the maximum number of h v taken over all vertices of G is called its spam, denoted by rmsn h , and the minimum value of rmsn h taking over all radio mean square labeling h of G is called the radio mean square number of G , denoted by rmsn G . In this study, we investigate the radio mean square numbers rmsn P n and rmsn C n for path and cycle, respectively. Then, we present an approximate algorithm to determine rmsn G for graph G . Finally, a new mathematical model to find the upper bound of rmsn G 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.

A new graph radio k-coloring algorithm

Due to the rapid growth in the use of wireless communication services and the corresponding scarcity and the high cost of radio spectrum bandwidth, Channel assignment problem (CAP) is becoming highly important. Radio [Formula: see text]-coloring of graphs is a variation of CAP. For a positive integer [Formula: see text], a radio [Formula: see text]-coloring of a simple connected graph [Formula: see text] is a mapping [Formula: see text] from the vertex set [Formula: see text] to the set [Formula: see text] of non-negative integers such that [Formula: see text] for each pair of distinct vertices [Formula: see text] and [Formula: see text] of [Formula: see text], where [Formula: see text] is the distance between [Formula: see text] and [Formula: see text] in [Formula: see text]. The span of a radio [Formula: see text]-coloring [Formula: see text], denoted by [Formula: see text], is defined as [Formula: see text] and the radio[Formula: see text]-chromatic number of [Formula: see text], denoted by [Formula: see text], is [Formula: see text] where the minimum is taken over all radio [Formula: see text]-coloring of [Formula: see text]. In this paper, we present two radio [Formula: see text]-coloring algorithms for general graphs which will produce radio [Formula: see text]-colorings with their spans. For an [Formula: see text]-vertex simple connected graph the time complexity of the both algorithm is of [Formula: see text]. Implementing these algorithms we get the exact value of [Formula: see text] for several graphs (for example, [Formula: see text], [Formula: see text], [Formula: see text], some circulant graph etc.) and many values of [Formula: see text], especially for [Formula: see text].

Traffic aware channel assignment with node stability in wireless mesh networks

In a wireless network the most challenging issue is Channel assignment. The channel assignment problem is codependent with the routing problem. We need to compute again the channel assignment as and when the traffic pattern changes. Anyways, previously followed channel assignment algorithms will assign channels from scratch. It will end up with an entirely dissimilar configuration of nodes; in turn it will disturb the action of the particular network. It takes little time to create links and to establish new channels. This time is significant in assigning links for wireless networks. This leads to channel reassignment. This algorithm considers the existing channel assignment and tries to go along with the new stream of traffic flow design in the finest possible way by changing the channel on a restricted number of radios. In order to provide node stability, we used entropy function. In this paper, we demonstrate a channel reallocation algorithm with node permanency and appraise its performance by using NS2. Experimental outcomes show that the node stability can progress the performance of network when compared with the earlier methods.