Radio Labelings of Lexicographic Product of Some Graphs

Aasi, Muhammad Shahbaz; Asif, Muhammad; Iqbal, Tanveer; Ibrahim, Muhammad

doi:https://doi.org/10.1155/2021/9177818

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Advances in Graph Labeling

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Research Article | Open Access

Volume 2021 | Article ID 9177818 | https://doi.org/10.1155/2021/9177818

Radio Labelings of Lexicographic Product of Some Graphs

Muhammad Shahbaz Aasi,¹Muhammad Asif,¹Tanveer Iqbal,¹and Muhammad Ibrahim¹

Academic Editor: Antonio Di Crescenzo

Received13 Sept 2021

Accepted02 Nov 2021

Published07 Dec 2021

Abstract

Labeling of graphs has defined many variations in the literature, e.g., graceful, harmonious, and radio labeling. Secrecy of data in data sciences and in information technology is very necessary as well as the accuracy of data transmission and different channel assignments is maintained. It enhances the graph terminologies for the computer programs. In this paper, we will discuss multidistance radio labeling used for channel assignment problems over wireless communication. A radio labeling is a one-to-one mapping satisfying the condition for any pair of vertices in . The span of labeling is the largest number that assigns to a vertex of a graph. Radio number of , denoted by , is the minimum span taken over all radio labelings of . In this article, we will find relations for radio number and radio mean number of a lexicographic product for certain families of graphs.

1. Introduction

The notion of graph labeling was first introduced in 1966 by Rosa in [1], and since then, many different graph labelings have been defined and studied. In the century, for studying the channel assignment problem, the term graph labeling was used where the transmitters are used as the vertices of the graph. Two vertices (transmitters) are said to be adjacent if they are sufficiently close to each other. A model of the channel assignment problem was provided by Hale [2] in 1980. Basic notions and definitions can be found in [3].

Let be a connected graph with vertex set and edge set . For any , let be the shortest length of the path between the vertices and . A distance-two labeling is a function with span having the maximum value such that for any , the following relations are satisfied:

In 1992, Griggs and Yeh [4] extensively studied about distance-two labeling.

An assignment of positive integers to the vertices of by of is said to be a radio -labeling if , where is an integer, . The span of labeling , denoted by , is the max . Radio number of , denoted by , is the minimum span taken over all radio labelings of . The radio -labeling number of is the minimum span among all radio -labelings of .

The study of radio -labelings was motivated by Chartrand et al. [5] where they found the radio -labeling number for paths. In [5], the lower and upper bounds were given for the radio -labeling number for paths which have been improved lately by Kchikech et al. [6]. The radio -labeling becomes a radio labeling, when . A radio labeling is a mapping from the vertices of the graph to some subsets of positive integers. The task of radio labeling is to assign to each station a positive smallest integer such that the interference in the nearest channel should be minimized. In 2001, multilevel distance labeling problem was introduced by Chartrand et al. [7].

A radio labeling is a one-to-one mapping satisfying the condition

In [8], multilevel distance (or radio) labeling for paths and cycles are determined by Liu and Zhu. Rahim et al. in [9] discussed and determined the radio number of Helm graphs. In [8], Liu et al. calculated the radio number of path graph. The radio numbers of hypercube graphs and square cycles have been computed by Khennoufa [10] and Liu et al. [11], respectively. In [12], Naseem et al. gave a lower bound for the radio number of edge-joint graphs. Adefokun and Ajayi [13] proved that for and even and that for even . Kim et al. [14] determined the radio numbers of with and with . Lower bound has been improved by Bantva [15] for the radio number of graphs which was earlier given by Das et al. in [16]. For more results, we have [17–21].

In [22], Ali et al. proposed a formula for finding a lower bound for , for graphs with small diameter. It is sometimes very useful to determine how many pairs with we can have. If there can be atmost ‘’ such pairs in a graph , then

In this paper, firstly, we determine the radio number and then radio mean number for the lexicographic product of path with path, path with cycle, and cycle with cycle. Finally, we present computer programs for finding such radio labelings of these families of graphs.

2. Applications

Labeling of graphs is one of the most popular parameters due to its diverse applications in real life. Radio labeling process proved as an efficient way of determining the time of communication for sensor networks. For giving valuable mathematical models, it has a wide scope of applications such as coding theory, electrical switchboards, circuit design, communication network addressing, channel assignment process, social networks, astronomy, demand and supply scenario, radar, database management, X-ray crystallography, and data security.

3. Lexicographic Product of Graphs

The lexicographic product was first studied by Hausdorff in 1914 [23]. The lexicographic product of two graphs and is denoted by which is a graph with (Figure 1)(1)The vertex set of the Cartesian product , and(2)Distinct vertices and are adjacent in iff(a), or(b) and .

4. Main Results

In this section, we discuss the radio labelings and compute the radio number for the lexicographic product of path with path and path with cycle for . Moreover, we also presented a computer program for computing the radio number of these families of graphs.

4.1. Results of Radio Labeling

Let be the path with vertices. The lexicographic product of with is isomorphic to graph . The radio number of paths is investigated by Liu et al. in [8] as stated in the following result.

Theorem 4.1 (see [8]). For any ,

We have a result for lower bound of for and .

Theorem 4.2. For all , .

Proof. In order to prove that the value stated above is a lower bound for the radio number, we will use the idea of distance-two labeling, i.e., expression 1.
The order of the graph is for and there exists , such pairs with labeling difference equals to 1. So, 3 implies that

Theorem 4.3. For all , .

Proof. The vertex set is partitioned in two disjoint sets and . Each partition is given as and . For , and . Define a mapping as follows: Claim: the mapping is a valid radio labeling. We must show that condition 2 for radio labeling holds for all pair of vertices . Case 1: suppose and are any two vertices in , then two subcases can be obtained. Case 1.1: let and be any two distinct vertices in , then and , ; therefore, and. . Also, we note that ; hence, . Case 1.2: let and be any two distinct vertices in , then and , ; therefore, and . Also, we note that ; hence, . Case 2: suppose and are any two vertices in , then two subcases can be obtained. Case 2.1: let and be any two distinct vertices in , then and , ; therefore, and . Also, we note that ; hence, . Case 2.2: let and be any two distinct vertices in , then and , ; therefore, and . Also, we note that ; hence, . Case 3: suppose and are any two vertices in , then two subcases can be obtained. Case 3.1: let and be any two distinct vertices in , then and , . therefore, and . Also, we note that ; hence, . Case 3.2: let and be any two distinct vertices in , then and , ; therefore, and . Also, we note that ; hence, .

Theorem 4.4.

Theorem 4.5. For all , .

Proof. The vertex set is partitioned in three disjoint sets , and . Each partition is further partitioned in two disjoint sets, i.e., , and . For , and . Define a mapping as follows: Claim: the mapping is a valid radio labeling. We must show that condition 2 for radio labeling holds for all pair of vertices . Case 1: suppose and are any two vertices in , then two subcases can be obtained. Case 1.1: let and be any two distinct vertices in , then and , ; therefore, and . Also, we note that ; hence, . Case 1.2: let and be any two distinct vertices in , then and , ; therefore, and . Also, we note that ; hence, . Case 2: suppose and are any two vertices in , then two subcases can be obtained. Case 2.1: let and be any two distinct vertices in , then and , ; therefore, and . Also, we note that ; hence, . Case 2.2: let and be any two distinct vertices in , then and , ; therefore, and . Also, we note that ; hence, . Case 3: suppose and are any two vertices in , then two subcases can be obtained. Case 3.1: let and be any two distinct vertices in , then and , ; therefore, and . Also, we note that ; hence, . Case 3.2: let and be any two distinct vertices in , then and , ; therefore, and . Also, we note that ; hence, .

4.2. Computing Radio Number of Lexicographic Product of Graphs by Using Computer Language

This computer code has been composed by using Python language. import numpy as np import math as mt def main(): m = int(input(‘m = Enter the number of vertices (either 2 or 3) = ‘)) n = int(input(‘n = Enter the number of vertices = ‘)) name3 = input(‘Type rnPP for lexico of two path graphs, Type rnPC for radio number of path and cycles, Type exist to quit the program: ‘) while name3 ! = ‘exit’: if name3 = = ‘rnPP’: print(‘Executing rnPP’) rnpp(n, m) elif name3 = = ‘rnPC’: print(‘Executing rnPC’) rnpc(n, m) else: print(‘Input error: Enter the correct input value.‘) name3 = input(‘Enter rnPP for lexico of two path graphs, rnPC for radio number of path and cycles, or exist to quit the program: ‘) def rnpc(n, m): if m = = 2: q1 = mt.ceil(n/2) l = np.zeros(n, dtype = int) r = np.zeros(n, dtype = int) for i in range(0, q1, 1): l[q1-1-i] = 2∗i r[q1-1-i] = (n+1) + 2∗i for j in range(1, n-q1+1, 1): l[n-j] = 2∗j-1 r[n-j] = n+2∗j for lc, rc in zip(l, r): print(lc, rc) elif m = = 3: q2 = mt.ceil(n/2) l = np.zeros(n, dtype = int) r = np.zeros(n, dtype = int) c = np.zeros(n, dtype = int) for i in range(0, q2, 1): l[q2-1-i] = (n+1) + 2∗i r[q2-1-i] = 2∗(n + i) + 1 c[q2-1-i] = 2∗i for j in range(1, n-q2+1, 1): l[n-j] = n + 2∗j r[n-j] = 2 ∗ (n + j) c[n-j] = 2 ∗ j-1 for lc, cc, rc in zip(l, c, r): print(lc, cc, rc) else: print(‘Try again! Enter either 2 or 3 for the value of m.‘) exit() def rnpp(n, m): if m = = 2: q1 = mt.ceil(n/2) l = np.zeros(n, dtype = int) r = np.zeros(n, dtype = int) for i in range(0, q1, 1): l[q1-1-i] = 2∗i + 1 r[q1-1-i] = (n+2) + 2∗i for j in range(1, n-q1+1, 1): l[n-j] = 2∗j r[n-j] = (n+1) + 2∗j for lc, rc in zip(l, r): print(lc, rc) elif m = = 3: q2 = mt.ceil(n/2) l = np.zeros(n, dtype = int) r = np.zeros(n, dtype = int) c = np.zeros(n, dtype = int) for i in range(0, q2, 1): l[q2-1-i] = (n+2) + 2∗i r[q2-1-i] = 2∗(n + i+1) c[q2-1-i] = 2∗i+1 for j in range(1, n-q2+1, 1): l[n-j] = n + 2∗j+1 r[n-j] = 2 ∗ (n + j) + 1 c[n-j] = 2 ∗ j for lc, cc, rc in zip(l, c, r): print(lc, cc, rc) else: print(‘Try again! Enter either 2 or 3 for the value of m.‘) exit() main()

5. Results of Radio Mean Labeling

Ponraj et al. [24] discussed the radio mean labeling. In this section, we discuss the radio mean labeling and compute the radio mean number for the lexicographic product of path with path and path with cycle for . Moreover, we also presented a computer program for computing the radio number of these families of graphs.

Definition 5.1. Radio mean labeling of a connected graph is a one-to-one map from the vertex set to the set of natural numbers such that for two distinct vertices and of ,The radio mean number of , denoted by , is the maximum number assigned to any vertex of . The radio mean number of is the minimum value of taken over all radio mean labeling of .

Theorem 5.2. For and , .

Proof. Let for and and . It is clear that . We define a vertex labeling as follows: for and . Now, we check the radio mean condition.for all . Case 1: the vertex labeling for the pair for a fixed , and , is given as and . Here, . So, . Case 2: check the pair for a fixed , and . , , and . So, . Case 3: check the pair for a fixed , and , for . , , and . So, .

5.1. Computing Radio Mean Number of Lexicographic Product of Graphs by Using Computer Language

This computer code has been composed by using Python language. import numpy as np print(‘Program to calculate the Radio Mean Labelling’) m = int(input(‘m = Enter the number of vertices (either 2 or 3) = ‘)) n = int(input(‘n = Enter the number of vertices )) if m = = 2: lt = np.zeros(n, dtype = int) rt = np.zeros(n, dtype = int) for j in range(1, m+1, 1): if j = = 1: for i in range(1, n+1, 1): lt[i-1] = m∗i - m + j. else: for i in range(1, n+1, 1): rt[i-1] = m∗i - m + j. for lc, rc in zip(lt, rt): print(lc, rc) elif m = = 3: lt = np.zeros(n, dtype = int) rt = np.zeros(n, dtype = int) ct = np.zeros(n, dtype = int) for j in range(1, m+1, 1): if j = = 1: for i in range(1, n+1, 1): lt[i-1] = m∗i - m + j elif j = = 2: for i in range(1, n+1, 1): ct[i-1] = m∗i - m + j else: for i in range(1, n+1, 1): rt[i-1] = m∗i - m + j for lc, cc, rc in zip(lt, ct, rt): print(lc, cc, rc) else: print(‘Error! The input value of m is either 2 or 3. Try again.‘)

6. Conclusion

In this paper, we have discussed the radio number and radio mean number of lexicographic product of graphs, namely, , , , and for . We also computed the exact value of radio number and radio mean number of these families. Moreover, in this paper, we have presented their computer codes and also two open problems for future work have been given.

7. Open Problems

(1)Determining the radio number of for .(2)Determining the radio mean number of for .

Data Availability

To support this study, no data were used.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright

Copyright © 2021 Muhammad Shahbaz Aasi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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