L(2,1) Labeling of Lollipop and Pendulum Graphs

Abstract

One of the topics in graph labeling is 𝐿(2,1) labeling which is an extension of graph labeling. Definition of 𝐿(2,1) labeling is a function that maps the set of vertices in the graph to non-negative integers such that every two vertices 𝑢, 𝑣 that have a distance one must have a label with a difference at least two. Furthermore, every two vertices 𝑢, 𝑣 that have a distance two must have a label with a difference at least one. This study discusses the 𝐿(2,1) labeling on a lollipop graph 𝐿𝑚,𝑛 with 𝑚 ≥ 3 and 𝑛 positive integers. The purpose of this study is to determine the minimum span value from the 𝐿(2,1) labeling on the lollipop graph 𝐿𝑚,𝑛 and we can symbolize 𝜆2,1(𝐿𝑚,𝑛) and to determine the minimum span value from the 𝐿(2,1) labeling on the pendulum graph. In addition, it also builds a simulation program for 𝐿(2,1) labeling lollipop graphs up to tremendous values of 𝑚 and 𝑛. In this paper, we obtained that the minimum span of a lollipop graph is 𝜆2,1(𝐿𝑚,𝑛) = 2𝑚 −2, and the minimum span of a pendulum graph, let 𝑃𝑛 𝑘 with 𝑘 ≥ 4 and 𝑛 ≥ 5, is 𝑘 + 1.