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### Upper Bound for the Radio Number of Some Families of Sunlet Graph

#### Abstract

Objectives: In this study, Radio Coloring is used to color the graphs. The objective of this article is to analyze the bounds of Line, Middle, Total and Central graphs of Sunlet graph. Methods/Analysis: Combinatorics is an important part of discrete mathematics that solves counting problems without actually enumerating all possible cases. Combinatorics has wide applications in Computer Science, especially in coding theory, analysis of algorithms and others. An equation that expresses an, the general term of the sequence {an} is called a recurrence relation. Using the generating function of a sequence and few coloring techniques we prove the results. Findings: The Problem of finding radio coloring with small or optimal k arises in the concept of radio frequency assignment. The radio chromatic score rs(G)17 of a radio coloring is the number of used colors. The number of colors used in a radio coloring with the minimum score is the radio chromatic rn(G) of G. The radio chromatic number of Sunlet graph Sn is 10 4 if n=3i, i=1,2,... 5 if n=3i+1, i=1,2,... 6 if n=3i+2, i=1,2,... and is improved to the radio chromatic number of Sunlet graph Sn is 5 if n is congruent to 1mod 3 4 if otherwise In this paper we improve the radio chromatic number of Sunlet graph Sn and obtain the radio number of Line, Middle, Total and Central graphs of Sunlet graph. Radio Coloring has wide range of significance because radio coloring has its applications in communication theory. The paper contributes the researches in the field of computer science and combinatorics. Applications: Radio coloring is of great significance because the frequency assignment problem is modeled as a graph coloring problem assuming transmitters as vertices and interference as adjacencies between two vertices.

#### Keywords

Sunlet Graph, Line Graph, Middle Graph, Radio Number, Total Graph and Central Graph.

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