TELKOMNIKA Telecommunication Computing Electronics and Control

Received Aug 29, 2020 Revised Dec 17, 2021 Accepted Dec 26, 2021 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 (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.


INTRODUCTION
In the field of communication engineering, the radio frequencies are commonly used in communication devices such as radio transmitters, computers, televisions, and mobile phones due to the fact that the frequency and energy of radio waves are very low.Researchers and engineers are working on optimizing the usage of the allotted bandwidth for a specified communication system due to the high cost of spectrum.In 1992, Griggs and Yeh [1] optimized the number of channels for the amplitude modulation (AM) radio stations in the stipulated bandwidth with the help of a graph labelling technique, known as distance two labelling.Motivated by the distance two labelling concept, Chartrand et al. [2] introduced in the early 21 st century the radio labelling concept for the frequency modulation (FM) radio stations.This type of channel allocation concerns with the maximum number of channels in a particular geographical area such that all the stations can receive the distinct frequencies.Since the distance between transmitters and their difference in frequency has played a vital role in assigning the maximum number of channels, the distance two labelling and radio labelling can be defined as follows: The distance two labelling, denoted by L(2,1)-labelling of a graph A, is a function, , from the vertex set V(A) to the set of all non-negative integers such that (, ) represents the distance between the vertices c and s in ; therefore, we have |() − ()|≥2 and TELKOMNIKA Telecommun Comput El Control
For a connected graph A, radio labelling is an injection, k, from the vertices of  to  such that d represents the diameter of a graph , the result, (, ) + |() − ()| ≥ 1 +  ∀ ,  ∈ (), is obtained.The radio numbers of  and A are represented by () and () which are the maximum number assigned to any vertex of  and the minimum value of () taken over all labellings k of , respectively.The following Figure 1 depicts the definition of radio number.
For the last two decades, several authors studied the radio labelling problem for general graphs and certain interconnection networks.The radio number of the total path of graphs were determined by Vaidya and Bantva [14].Cada et al. [15] obtained the radio number of distance graphs.The same problem for trees was studied by Liu [16].Kim et al. [17] presented the product of graphs namely   ( ≥ 4) and   ( ≥ 2).Bharati and Yenoke [18] determined both upper and lower bounds for the hexagonal mesh as (3 2 − 4 − 1) + 3 and , respectively.Bantva [19] slightly improved the lower bound that was established in [20].In addition, Yenoke et al. [21] proved that   ( (, )) ≤ ( − 2)(4 2 − 9 + 8) + 2( − 1) 2 + ( + 1), where  (, ) is the enhanced mesh,  ≥ 4.This paper is divided as follows: In section 2, we discuss the methodology of our research work.In section 3, our main results are obtained by studying the bounds for the L(2,1)-labelling number and radio number of the general tree dendrimer  , .Our research work is concluded in section 4.

RESEARCH METHOD
The author studied in [18], [21] the same problem for the networks that contains the number of vertices in the n th dimension as 3 2 − 3 + 1 and  2 , respectively.In addition, the authors in [14], [15], [17] studied the same problem for the graphs with  vertices.Since the vertices have been increased in terms of the high order for a network, especially of order   ; therefore, finding a good solution is very complicated.The authors are trying to find a solution for such networks.However, a good bound is obtained in this paper for the tree dendrimer chemical network which grows in the order of generation.Further, most of the networks were studied separately for the L(2,1) labelling or radio labelling.Due to the exponential growth of communication technology, we are today in need to estimate the lower and upper bounds for the graphs that are growing in higher order to compete with the consumers' demand.By taking this into our account and according to the best of our knowledge, for the first time ever, we have estimated in this research work the bounds for both L(2,1)-labelling and radio labelling numbers for an   (number of vertices) expanding chemical structure, known as nanaostar tree dendrimer.This research study provides a detailed analysis of the growth of such graph in terms of diameter and vertices for ,  > 2, and its bounds have been obtained separately for (2,1)-labelling number and radio labelling number.Therefore, all obtained results in this study are novel and worthy.

Nanostar tree dendrimer
Nanostar is a star-looking type of nanoparticle that contains a spherical core with many branches.Dendrimers have very complex chemical structures and hyper-branched macromolecules with a star-shaped architecture.In addition, dendrimers are classified by a generation which represents the repeated branching cycles number that are performed during its synthesis.The structure of these materials has a huge impact on the physical and chemical properties of dendrimers due to the uniqueness of dendrimers' behavior which makes them very suitable for various biomedical and industrial applications [22]- [24].Yang and Xia [24] defined a tree dendrimer graph, denoted by  , , as follows: The center vertex of the graph  , is represented by  1 0 which is a  −regular graph except the pendant vertices.In addition, the distance from the center vertex  1 0 to every pendant vertex is exactly .Moreover,  signifies here the  ℎ generation of the tree dendrimer.The diameter and radius of a tree dendrimer graph are 2 and , respectively.
In this research work, we have named the  generation vertices of the tree dendrimer  , as follows: First, we name the  vertices in the first generation which are adjacent to the center vertex  1 0 as
Proof: By defining a mapping : ( 2, ) → , we have the following:  In addition, the vertices  (−1) ⌋ attains the maximum value 3 + 2, which implies that  2,1 () + 1 ≤ 3 + 2. Thus,  2,1 () ≤ 3 + 1. Remark 1: The branches of the tree dendrimer which are connected to the center vertex  1 0 by a single edge are called the main branches of the tree dendrimer graph.We denote the  main branches in  , as   ,  = 1,2 … .Lemma 3.1: Let  , be a tree dendrimer graph of  generations with each vertex of degree  expects the pendant vertices, then the number of vertices in each main branch   (1 ≤  ≤ ) is Proof: From the construction of the tree dendrimer, the first generation contains  − 1 vertices.Since the root vertex of a main branch is a vertex of first generation, there is only a single vertex in the first generation.Therefore, the number of vertices in a second generation is  − 1.In general, the number of vertices in the  ℎ generation is ( − 1) −1 .Hence, the total number of vertices in a main branch is calculated as follows: .Hence, the radio number of  , (,  > 2) satisfies ( , ) ≤  + (2 − 1) + 1 + ∑ (2 − −1 =1 1)(( − 1) −−1 ))( − 2) + (2 − 1)() −−1 − 1, whenever  ≥ 2 − 3. Hence, the theorem is proven.Next, we determine the lower bound for the radio number of  , (,  > 2) by using the following theorem which was proven by Bharati and Yenoke [18]: Theorem 3.5 (As Theorem 2 in [18]): Let A be a simple connected graph of order m.Let  0 ,  1 …   be the number of vertices that have eccentricities  0 ,  1 …   , where () =  =  0 >  1 > ⋯ >   = ().Then, we obtain the following: