Research Article Radio Labelings of Lexicographic Product of Some Graphs Muhammad Shahbaz Aasi , Muhammad Asif , Tanveer Iqbal , and Muhammad Ibrahim Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan CorrespondenceshouldbeaddressedtoMuhammadShahbazAasi;mshahbazasi@gmail.com Received 13 September 2021; Accepted 2 November 2021; Published 7 December 2021 AcademicEditor:AntonioDiCrescenzo Copyright © 2021 Muhammad Shahbaz Aasi et al. is is an open access article distributed under the Creative Commons AttributionLicense,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkis properly cited. Labelingofgraphshasdefinedmanyvariationsintheliterature,e.g.,graceful,harmonious,andradiolabeling.Secrecyofdatain datasciencesandininformationtechnologyisverynecessaryaswellastheaccuracyofdatatransmissionanddifferentchannel assignments is maintained. It enhances the graph terminologies for the computer programs. In this paper, we will discuss multidistanceradiolabelingusedforchannelassignmentproblemsoverwirelesscommunication.Aradiolabelingisaone-to-one mapping ℘: V(G) ⟶ Z + satisfyingthecondition |℘(μ)− ℘(μ′ )| ≥ diam(G)+ 1 − d(μ, μ′ ): μ, μ′ ∈ V(G) foranypairofvertices μ, μ′ in G.espanoflabeling ℘ isthelargestnumberthat ℘ assignstoavertexofagraph.Radionumberof G,denotedby rn(G), istheminimumspantakenoverallradiolabelingsof G.Inthisarticle,wewillfindrelationsforradionumberandradiomean number of a lexicographic product for certain families of graphs. 1. Introduction enotionofgraphlabelingwasfirstintroducedin1966by Rosa in [1], and since then, many different graph labelings have been defined and studied. In the 19 th century, for studying the channel assignment problem, the term graph labeling was used where the transmitters are used as the verticesofthegraph.Twovertices(transmitters)aresaidto be adjacent if they are sufficiently close to each other. A model of the channel assignment problem was provided by Hale[2]in1980.Basicnotionsanddefinitionscanbefound in [3]. Let G �(V(G),E(G)) beaconnectedgraphwithvertex set V(G) and edge set E(G). For any μ, μ ′ ∈ V(G), let d(μ, μ ′ ) be the shortest length of the path between the vertices μ and μ ′ . A distance-two labeling is a function ℘: V(G) ⟶ 1, 2, 3, ... ,k { } with span k having the maxi- mum value k such that for any μ, μ ′ ∈ V(G), μ ≠ μ ′ , the following relations are satisfied: |℘(μ)− ℘ μ ′ ( | ≥ 2, if d μ, μ ′ (  � 1 1, if d μ, μ ′ (  � 2 . ⎧⎨ ⎩ (1) In 1992, Griggs and Yeh [4] extensively studied about distance-two labeling. Anassignmentofpositiveintegerstotheverticesof G by ℘ of G is said to be a radio k-labeling if |℘(μ)− ℘(μ ′ )| ≥ k + 1 − d(μ, μ ′ ),where k isaninteger, k ≥ 1. e span of labeling ℘, denoted by sp(℘), is the max |℘(μ)− ℘(μ ′ )|: μ, μ ′ ∈ V(G)  .Radionumberof G,denoted by rn(G),istheminimumspantakenoverallradiolabelings of G.eradio k-labelingnumberof G istheminimumspan among all radio k-labelings of G. e study of radio k-labelings was motivated by Char- trandetal.[5]wheretheyfoundtheradio k-labelingnumber forpaths.In[5],thelowerandupperboundsweregivenfor the radio k-labeling number for paths which have been improved lately by Kchikech et al. [6]. e radio k-labeling becomes a radio labeling, when k � diam(G). A radio la- beling is a mapping from the vertices of the graph to some subsets of positive integers. e task of radio labeling is to assigntoeachstationapositivesmallestintegersuchthatthe interferenceinthenearestchannelshouldbeminimized.In 2001, multilevel distance labeling problem was introduced by Chartrand et al. [7]. Hindawi Journal of Mathematics Volume 2021, Article ID 9177818, 6 pages https://doi.org/10.1155/2021/9177818