TELKOMNIKA Telecommunication, Computing, Electronics and Control
The bounds for the distance two labelling and radio labelling of nanostar tree dendrimer

TELKOMNIKA Telecommunication, Computing, Electronics and Control
The bounds for the distance two labelling and radio labelling of nanostar tree dendrimer

Distance-two-labelling, Labelling, Radio labelling, Radio number

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 non-
negative 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 (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.

Kins Yenoke, Mohammed K. A. Kaabar

DOI: 10.12928/TELKOMNIKA.v20i1.20404

Universitas Ahmad Dahlan

February 2022

Sri Wahyuni

ISSN: 1693-6930

http://journal.uad.ac.id/index.php/TELKOMNIKA

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TELKOMNIKA Telecommunication, Computing, Electronics and Control