The bounds for the distance two labelling and radio labelling of nanostar tree dendrimer

Abstract

The distance two labelling and radio labelling problems are applicable to find the optimal frequency assignments on AM and FM radio stations. The distance two labelling, known as L(2,1)-labelling of a graph A, can be defined as a function, 𝑘, from the vertex set V(A) to the set of all nonnegative integers such that 𝑑(𝑐, 𝑠) represents the distance between the vertices c and s in 𝐴 where the absolute values of the difference between 𝑘(𝑐) and 𝑘(𝑠) are greater than or equal to both 2 and 1 if 𝑑(𝑐, 𝑠)=1 and 𝑑(𝑐, 𝑠) = 2, respectively. The L(2,1)-labelling number of 𝐴, denoted by 𝜆2,1 (𝐴), can be defined as the smallest number j such that there is an 𝐿(2,1) −labeling with maximum label j. A radio labelling of a connected graph A is an injection k from the vertices of 𝐴 to 𝑁 such that 𝑑(𝑐, 𝑠) + |𝑘(𝑐) − 𝑘(𝑠)| ≥ 1 + 𝑑 ∀ 𝑐, 𝑠 ∈ 𝑉(𝐴), where 𝑑 represents the diameter of graph 𝐴. The radio numbers of 𝑘 and A are represented by 𝑟𝑛(𝑘) and 𝑟𝑛(𝐴) which are the maximum number assigned to any vertex of 𝐴 and the minimum value of 𝑟𝑛(𝑘) taken over all labellings k of 𝐴, respectively. Our main goal is to obtain the bounds for the distance two labelling and radio labelling of nanostar tree dendrimers.