Radio antipodal number of honeycomb derived networks

Abstract

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 $$\tau$$ τ from the vertex set of a graph T to the set of natural numbers such that the condition $$d(\alpha ,\eta )+\mid \tau (\alpha )-\tau (\eta )\mid \ge diam(T)$$ d ( α , η ) + ∣ τ ( α ) - τ ( η ) ∣ ≥ d i a m ( T ) , is satisfied. The span of the antipodal labeling $$\tau$$ τ is the maximum label allotted in a graph and is given by $$sp(\tau )=max\{\mid \tau (\alpha )-\tau (\eta )\mid :\alpha ,\eta \in V(T)\}$$ s p ( τ ) = m a x { ∣ 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.

Radio Labeling of Supersub—Division of Path Graphs

“…These concepts include CAP, radio labeling, radio number, and frequency assignment. Authors [10] examined a comprehensive overview of graph labeling and the authors [11] discussed an assignment of radio numbers to triangular and rhombic honeycomb networks. In addition, the study of the radio and antipodal number for many different types of graphs has been the subject of intellectual research by several authors ( [12]; [13]; [14]; [15]; [16]; [17]; [18]; [19]; [20]; [21]; [22]).…”