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The need for smart and sustainable communication systems has led to the development of mobile communication networks. In turn, the vast functionalities of the global system of mobile communication (GSM) have resulted in a growing number of subscribers. As the number of users increases, the need for efficient and effective planning of the "limited"...
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... An efficient allocation algorithm must be developed to effectively manage network constraints and mitigate sporadic communications [42]. In addition, the main objective is to manage the constraints associated with wireless communications, especially in high-density areas where multiple wireless networks coexist ( [43]; [44]). ...
Motivated by the channel assignment problem, we study the radio labeling of graphs. The radio labeling problem is an important topic in discrete mathematics due to its diverse applications, e.g., frequency assignment in mobile communication systems, signal processing, circuit and sensor network design, etc. A graph labeling problem is an assignment of labels to the vertices or edges (or both) of a graph G that satisfy a mathematical constraint. Radio labeling, a vertex labeling of graphs with non-negative integers, finds an important application in the study of radio channel assignment problems. The maximum label used in a radio labeling is called its span, and the smallest possible span of a radio labeling is called the radio number of a graph. In this area, Liu and Zhu [1] provided important results by computing the exact values of
rn(G)
for paths and cycles when
k
is equal to the diameter for certain cases. In this paper, we determine the radio number
rn(G)
of
G
where
G
is the supersub–division of a path
P<sub>n</sub>
with
n
≥ 3 vertices and a complete bipartite graph
K
<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2,α</sub>
with α ∈ N.
... Case (2): Suppose the nodes lie in different levels and d(α, η) = 1. ...
The communication in a wireless network mainly depends on the frequencies or channels assigned to them. The channels must be assigned to all the transmitters in the network without interference for effective communication. This problem is said to be a channel (frequency) assignment problem (CAP). With the limited availability of channels, CAP has become a challenging problem. This problem is modeled as a graph, where each transmitter is represented by a vertex, and two vertices are adjacent when their corresponding transmitters are close. The labelling technique in graph theory has played an important role in solving CAP, thereby the time and cost will be saved. In radio antipodal labeling, the channels were reused again for the antipodal vertices. It will reduce the usage of the number of channels, with minimum interference. Hence it is a better labeling compared to other labelings. It is a mapping τ from the vertex set of a graph T to the set of natural numbers such that the condition d(α,η)+∣τ(α)-τ(η)∣≥diam(T), is satisfied. The span of the antipodal labeling τ is the maximum label allotted in a graph and is given by sp(τ)=max{∣τ(α)-τ(η)∣:α,η∈V(T)}. The lowest value of all the spans of the antipodal labeling of graph T is said to be radio antipodal number. It is denoted by an(T). The value of the minimum span gives the bandwidth or spectrum of the channels. The honeycomb network plays an important role in communication engineering because of its structure. In this paper, the bounds of the antipodal number of honeycomb derived networks—triangular and rhombic honeycomb were obtained and represented graphically. These bounds give the optimum number of channels (bandwidth) needed for these honeycomb derived networks for effective communication without interference.
