Radio Labelings of Lexicographic Product of Some Graphs

Labeling of graphs has defined many variations in the literature, e.g., graceful, harmonious, and radio labeling. Secrecy of data in data sciences and in information technology is very necessary as well as the accuracy of data transmission and different channel assignments is maintained. It enhances the graph terminologies for the computer programs. In this paper, we will discuss multidistance radio labeling used for channel assignment problems over wireless communication. A radio labeling is a one-to-one mapping ℘ : V G ⟶ ℤ + satisfying the condition | ℘ μ − ℘ μ ′ | ≥ diam G + 1 − d μ , μ ′ : μ , μ ′ ∈ V G for any pair of vertices μ , μ ′ in G . The span of labeling ℘ is the largest number that ℘ assigns to a vertex of a graph. Radio number of G , denoted by r n G , is the minimum span taken over all radio labelings of G . In this article, we will find relations for radio number and radio mean number of a lexicographic product for certain families of graphs.

Radial Radio Number of Hexagonal and Its Derived Networks

A mapping ℸ : V G ⟶ N ∪ 0 for a connected graph G = V , E is called a radial radio labelling if it satisfies the inequality ℸ x − ℸ y + d x , y ≥ rad G + 1 ∀ x , y ∈ V G , where rad G is the radius of the graph G . The radial radio number of ℸ denoted by r r ℸ is the maximum number mapped under ℸ . The radial radio number of G denoted by r r G is equal to min { r r ℸ / ℸ is a radial radio labelling of G }.

Zagreb Radio Indices and Coindices

Zagreb radio index or coindex of a graph is defined based on the optimal radio labelling of the graph, where optimal refers to the labeling with a span equal to the radio number of the graph. We have defined the first and second Zagreb radio coindex, third Zagreb radio index, and coindex and calculated its value for some famous graphs. Furthermore, we established a certain relationship between Zagreb radio indices and the coindices of graphs.

Radio Number Associated with Zero Divisor Graph

Radio antennas use different frequency bands of Electromagnetic (EM) Spectrum for switching signals in the forms of radio waves. Regulatory authorities issue a unique number (unique identifying call sign) to each radio center, that must be used in all transmissions. Each radio center propagates channels to the two nearer radio centers so they must use distinctive numbers to avoid interruption. The task of effectively apportioning channels to transmitters is known as the Channel Assignment (CA) problem. CA Problem is discussed under the topic of graph coloring by mathematicians. The radio number of a graph can be used in many parts of the field communication. In this paper, we determined the radio number of zero-divisor graphs Γ(Zp2×Zq2) for p,q prime numbers.