Abstract
Objectives: In communication engineering, the assignment of channels or frequencies to different transmitters in a communication network without interference is an important problem. Finding the span for such an assignment is a challenging task. The objective of this study is to find the span of quadrilateral snake families. Method: The solution to the channel assignment problem can be found out by modeling the communication network as a graph, where the transmitters are represented by nodes and connectivity between transmitters are given by edges. The labeling technique in graph theory is very useful to solve this problem. Let G=(V;E) be a graph with vertex set V, edge set E. Let u;v 2V(G). The radio antipodal mean labeling of a graph G is a function f that assigns to each vertex u, a non-negative integer f (u) such that f (u) ̸= f (v) if d(u;v) < diam(G) and d(u;v)+⌈f (u)+ f (v)2⌉ diam(G) , where d(u;v) represents the shortest distance between any pair of vertices u and v of G and diam(G) is the diameter of G. The radio antipodal mean number of f, is the maximum number assigned to any vertex of G and is denoted by ramn( f ). The radio antipodal mean number of G, denoted by ramn(G) is the minimum value of ramn( f ) taken over all antipodal mean labeling f of G. Findings: In this study, we have obtained the bounds of radio antipodal mean number of quadrilateral snake families. Novelty: The radio antipodal mean number of quadrilateral snake families was not studied so far. Hence, the establishment of the bounds for radio mean number of quadrilateral snake families will motivate many researchers to study the radio antipodal mean number of other communication networks. Keywords: Radio antipodal mean labeling; quadrilateral snake; alternate quadrilateral snake; double quadrilateral snake; double alternate quadrilateral snake
Highlights
The radio labeling technique has a lot of application in communication engineering
The radio antipodal labeling condition was modified by Xavier and Thivyarathi (12) in 2018 and introduced a new graph labeling technique called radio antipodal mean labeling
The radio antipodal mean labeling of a graph G is a fun⌈ction f tha⌉t assigns to each vertex u, a non-negative integer f (u) such that f (u) ̸= f (v) if d(u, v) < diam(G) and d(u, v) +
Summary
The radio labeling technique has a lot of application in communication engineering. It is mainly used to assign channels or frequencies to different radio stations. Yeh (7) devised this problem as a distance 2 labeling or L(2, 1) labeling It is defined as follows: Given a real number d > 0, an Ld(2, 1) - labeling of G is a non-negative real-valued function f : V (G) → (0, ∞) such that, whenever x and y are two adjacent vertices in V , . The work of Jerrold et al motivated Chartrand et al(8) to introduce a new graph labeling technique called radio labeling. In 2015, Ponraj et al(11) redefined the radio labeling condition and introduced a new graph labeling technique called radio mean labeling. The radio antipodal labeling condition was modified by Xavier and Thivyarathi (12) in 2018 and introduced a new graph labeling technique called radio antipodal mean labeling. The bounds of radio antipodal mean number of quadrilateral snake families have been obtained
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