# Radial Radio Number of Hexagonal and Its Derived Networks

Authors: Kins Yenoke; Mohammed K. A. Kaabar; Mohammed M. Ali Al-Shamiri; R. C. Thivyarathi
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

A mappingℸ:VG⟶N∪0 for a connected graph G=V,E is called a radial radio labelling if it satisfies the inequality ℸx−ℸy+dx,y≥radG+1∀x,y∈VG, where radG is the radius of the graph G. The radial radio number of ℸ denoted by rrℸ is the maximum number mapped under ℸ. The radial radio number of G denoted by rrG is equal to min {rrℸ/ℸ is a radial radio labelling of G}.


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## Body

## 1. Introduction

In this twenty-first century, our day-to-day life is pervaded by the communication devices which are functioning with the help of electromagnetic waves. By designating a portion of the electromagnetic spectrum that has wavelengths ranging from 1 mm to 100 kms, or equivalently, frequencies from 300 GHz to 3 kHz are called radio waves which are used in the field of communication engineering. Due to the high cost of the spectrum, maximizing the number of channels in a predefined bandwidth gives a huge profit to the country. Hence, the graph labelling concepts play a vital role in maximizing such an optimization channel assignment problem. This channel assignment problem inspired Hale [1] in 1980 to introduce the graph-theoretic concept using graph labelling. Chartrand et al. [2] were motivated by these concepts in 2001 and introduced a new channel assignment problem called the radio labelling problem which is used to allot the maximum number of channels for the frequency modulation radio stations. A radio labelling of a connected graph G is an injection ℸ:VG⟶N such that dx,y+ℸx−ℸy≥1+diamG∀x,y∈VG. The radio number of ℸ, denoted by rnℸ, is the maximum number assigned to any vertex of G. The radio number of G, denoted by rnG, is the minimum value of rnℸ taken over all labelings ℸ of G. Computing such a problem for graphs with diameter two itself is NP-hard [3]. For the past two decades, several authors have studied the radio number problem for general graphs and interconnection networks. In the recent years, researchers have introduced the variations of radio labelling and called them as a different labelling either by changing ℸx–ℸy as ℸx+ℸy/2, ℸxℸy, ℸxℸy/2, and 2ℸxℸy/ℸx+ℸy or diamG as diamG−1, radG, etc.Motivated by the radio labelling problem, in order to increase the number of channels by splitting the given geographical area into two subregions, Selvam et al. [4] brought in the radial radio labelling concept in 2017. A mapping ℸ:VG⟶N∪0 for a connected graph G=V,E is called a radial radio labelling if this satisfies the inequality ℸx−ℸy+dx,y≥radG+1∀x,y∈VG, where radG is the radius of the graph G. The radial radio number of ℸ denoted by rrℸ is the maximum number mapped under ℸ. The radial radio number of G denoted by rrG is equal to min {rrℸ/ℸ is a radial radio labelling of G}. It is obvious from the definitions that, for any connected graph G, rnG≥rrG. However, the radial radio number is reduced to the radio number for any self-centered graphs, which is for the graphs that satisfy radG=diamG. For example, the radio number and radial radio number for the complete graphs Kn and complete bipartite graphs Km,n are nandm+n+1, respectively. Hence, rnG>rrG for any graph G which is not self-centered. Selvam et al. [4] proved that rrG≥n for any self-centered graph G. In addition, Selvam et al. [5] proved few results connecting the clique number ω and rrG as follows: (i) ωG≤rrG, (ii) for m≥1,∃ a graph G which satisfies rrG=m+ωandω=3, and (iii) ∃ a graph G with rrG=ω+1,wheneverω≥4. Yenoke [6] determined the upper bounds for the radial radio number of certain uniform cyclic and split graphs. Moreover, Arputha Jose and Giridharan [7] proved that rrMTn≤2n+1 and rrDn≤2n+2, where MTn is the Mongolian tent and Dn is the diamond graph. This research article highlights the newly defined enhanced hexagonal difference hexagonal network. Also, the radial radio number for HXη, HDNη, and EDHη was resolved.


## 2. Hexagonal and Its Derived Networks

In 2D geometry, the hexagonal network is the triangular tessellation of the Euclidean plane, and this was broadly analysed in [8–10]. In the graph theoretical approach, a hexagonal network of dimension η is denoted by HXη which contains 6η−2 vertices of degree 4, 6 corner vertices of degree 3, and 3η2−9η+7 vertices of degree 6. It was identified that each side of a network is equal to the dimension η. Also, ∃ a unique centre vertex at a distance η−1 from the corner vertices. Therefore, diamHXη and radHXη are 2η−2 and η−1, respectively. In addition, it has 9η2−15η+6 edges and 3η2−3η+1 vertices. See Figure 1(a).Figure 1
HXη, HDNη, and EDHη for η=6.
(a)(b)(c)Manuel et al. [11] placed a vertex in the face of each triangle and then joined it by the corresponding three vertices of the triangle and derived an architecture from the hexagonal network called an enhanced hexagonal network, and it is denoted by HDNη. In addition, EHDNη=9η2−15η+7 and VHDNη=27η2−51η+24. See Figure 1(b). Also, the diameter and radius are the same as in HXη. This paper studies a new network named enhanced hexagonal difference hexagonal network which is obtained from the enhanced hexagonal network by removing all the edges of HXη from HDNη. That is, EDHη=HDNη−EHXη. It is denoted by EDHη. The number of vertices and edges in EDHη is 9η2−15η+7 and 27η2−51η+24−9η2−15η+6=18η2−36η+18, respectively. See Figure 1(c). Furthermore, the diameter and the radius for EDHη are 4η−1 and 2η−1, respectively.Even though the vertices of the three axesα,β, and γ of HXη are already defined in the literature, for the requirement of the proof, we have renamed the vertices of the vertical lines β−lines from the left most top to the right most bottom as y11,y21,…,yκ1,y12,y22,…,yη+12,…,y1η+1,y2η+1,…,y2κ−1η+1,…,y12η−1,y22η−1,…,yη2η−1. Furthermore, the face vertices of HDNη are named in the same manner from the left most top to the right most bottom as x11,x21,…,xη−11,x12,x22,…,xη2,x13,x23,…,xη3,x14,x24,…,xη+14,…,x12η−2,x22η−2,…,x2η−22η−2, x12η−1,x22η−1,…,x2η−22η−1,…,x14η−1,x24η−1,…,xη−14η−1. See Figure 2(a).Figure 2
Naming of vertices inHDN3 and a radial radio labelling of HDN4 which attains the bound.
(a)(b)


## 3. Radial Radio Number ofHXη, HDNη, and EDHη

In this section, we have determined the upper bounds for the radial radio number of the hexagonal network, the enhanced hexagonal network, and the newly defined networkEDHη.Theorem 1.
LetHXη be a hexagonal network of dimension η; then, the radial radio number of HXη satisfies rrHXη≤η−1η2−η−1+1.Proof.
First, we partition the vertex setVHXκ into five disjoint sets W1=yσℷ/σ=1,2,…,η,ℷ=1,2,…,η−1, W2=yση−1+ℷ/σ=1,2,…,η,ℷ=1,2,…,η−1, W3=yη+σℷ+1/σ=1,2,…,ℷ,ℷ=1,2,…,η−2, W4=yη+ση−1+ℷ/σ=1,2,…,η−ℷ,ℷ=1,2,…,η−1, and W5=yσ2η−1/σ=1,2,…,η.
Define a mappingℸ:VHXη⟶N∪0 as follows:(1)ℸyσℷ=η−1σ−1+ℷ−1η−1η,σ=1,2,…,η,ℷ=1,2,…,η−1,ℸyση−1+ℷ=η−1σ−1+η−1ηℷ−1+1,σ=1,2,…,η,ℷ=1,2,…,η−1,ℸyη+σℷ+1=η−1σ+ℷη−1,σ=1,2,…,ℷ,ℷ=1,2,…,η−2,ℸyη+ση−1+ℷ=1+η−1−1+σ+ηℷ,σ=1,2,…,η−ℷ,ℷ=1,2,…,η−1,ℸyσ2η−1=η−1σ−1,σ=1,2,…,η.
See Figure2(b).
Next, we claim thatℸ satisfies the radial radio labelling condition. Since radHXη=η−1, we must verify that ℸx−ℸy+dx,y≥η−1+1=η∀x,y∈VHXη.
Letx,y∈HXη.Case 1: ifx,y∈W1, then x=yrs and y=ypq, where 1≤r,p≤η, 1≤s,andq≤η−1. Therefore, ℸx=η−1r−1+η−1ηs−1, ℸy=η−1p−1+η−1ηq−1, and dx,y≥1. If s=qandr≠p, then(2)ℸx−ℸy+dx,y≥η−1r−1+η−1η,s−1−η−1p−1+η−1ηq−1+1≥η−1r−p+1≥η.Ifr=p and s≠q, then ℸxℸy+dx,y≥η−1ηs−q+1>η. Otherwise, the condition is obviously true.Case 2: ifx,y∈W2, then x=yrη+s and y=ypη−1+q, where 1≤r,p≤η, 1≤s,andq≤η−1. Here, dx,y≥1. Furthermore, ℸxℸy=η−1r−1+η−1ηs−1+1−η−1p−1+η−1ηq−1+1. If both r≠p and s≠q, then there is nothing to verify. If s=q and r≠p or s≠q and r=p, then as in the previous case, ℸx−ℸy+dx,y≥η−1r−p+1≥η and ℸx−ℸy+dx,y≥η−1ηs−q+1>η.Case 3: supposex,y∈W3; then, dx,y≥1,ℸx=η−1r+ηs−1, and ℸy=η−1p+ηq−1, where 1≤r,p,s,q≤η−2. If either r≠p or s≠q, then ℸxℸy+dx,y≥η−1r−p+1≥η or ℸxℸy+dx,y≥η−1ηs−ηq+1>η.Case 4: ifx,y∈W4, thenℸx=ℸyη+rη−1+s=η−1r+ηs−1+1 and ℸy=ℸyη+pη−1+q=η−1p+ηq−1+1, where 1≤r,p,s,q≤η−1. Moreover, dx,y≥1. Hence, as in Case 3, we can verify that ℸx−ℸy+dx,y≥η.Case 5: supposex and y lie in the set W5; then,x=ys2η−1andy=yt2η−1,where1≤s≠t≤η. Hence, the values of x and yunderℸareη−1s−1andη−1t−1,respectively. In addition, dx,y≥1. Hence, ℸx−ℸy+dx,y≥η−1s−t+1≥η since s≠t.Case 6: ifx∈W1 and y∈W2, then x=yrs and y=ypη−1+q, where 1≤r,p≤η, 1≤s,andq≤η−1. Therefore, ℸxandℸyvalues areη−1r−1+η−1ηs−1 and η−1p−1+η−1ηq−1+1, respectively. Since x and y are at a minimum distance, η−1andℸx−ℸy≥1, we get ℸx−ℸy+dx,y≥1+η−1≥η.Case 7: supposex∈W1 and y∈W3; then, ℸyrs=η−1r−1+η−1ηs−1andℸyη+pq+1=η−1p+ηq+−1, where 1≤r≤η, 1≤s≤η−1, and 1≤p,q≤η−2. If r=pands=q, then dx,y=η. Again, if either r≠p or s≠q, then ℸxℸy≥η−1 and dx,y≥1. Hence, for the above possibilities, ℸx−ℸy+dx,y≥η.Case 8: ifx∈W1 and y∈W4, then the verification is the same as Case 6 since ℸx−ℸy≥1 and dx,y≥η−1.Case 9: ifx∈W1 and y∈W5, then x=yrs and y=yp2η−1,where1≤r,p≤η,and1≤s≤η−1. Since dx,y≥η+1, the condition is trivially satisfied.Case 10: ifx∈W2 and y∈W3, then x=yrη−1+s and y=yη+pq+1, where 1≤r≤η, 1≤p,q≤η−2,and1≤s≤η−1. Therefore, ℸx=η−1r−1+η−1s−1η+1,ℸy=η−1p+ηq+−1, and dx,y≥η−1. Hence, ℸx−ℸy+dx,y≥η−1r−1+η−1ηs−1+1η−1p+ηq+−1+η−1≥η.Case 11: ifx∈W2 and y∈W4, then the verification is similar to Case 7.Case 12: ifx∈W2 and y∈W5, then ℸx−ℸy=η−1r−1+η−1ηs−11−η−1t−1ℸxℸy=η−1r−1+η−1ηs−11−η−1t−1. where x=ytη−1+sandy=yt2η−1,1≤r,t≤η,1≤s≤η−1.Ifs=1, then dx,y=η−1. Therefore, ℸx−ℸy+dx,y≥η−1r−1+1−η−1t−1+η−1≥η.Ifs>1, then dx,y≥1, and hence, ℸx−ℸy+dx,y≥η−1r−1+η−1η+1−η−1t−1≥ηη−1+1>η.Case 13: assumex∈W3 and y∈W4; then, ℸx=yη+rs+1=η−1r+ηs−1 and ℸy=yη+pη−1+q=η−1p+η−1+1, where 1≤p,q≤η−1,1≤r,s≤η−2. If p=r+1, then dx,y≥η−1 and ℸx−ℸy≥1; otherwise, dx,y≥1 and ℸx−ℸy≥ηη−1. Hence, we get ℸx−ℸy+dx,y≥η.Case 14: letx∈W3 and y∈W5; then, ℸxℸy=η−1r+ηs−1−η−1t−1, where 1≤p,q≤η−2,1≤t≤η. If p=r+1, then dx,y≥η−1 and ℸxℸy≥1; otherwise, dx,y≥1 and ℸxℸy≥ηη−1. Hence, we get ℸxℸy+dx,y≥η.Case 15: letx∈W4 and y∈W5; then, the verification is the same as in Case 12.
Thus,ℸx−ℸy+dx,y≥η∀x,y∈VHXη. Hence, ℸ is a valid radial radio labelling, and the vertex yη2η−2 attains the maximum value, η−1η−1+η−1ηη−1−1+1=η−1η−1+ηη−2+1=η−1η2−η−1+1 Therefore, rrHXη≤η−1η2−η−1+1.Theorem 2.
LetHDNη be an enhanced hexagonal network of dimension κ; then, the radial radio number of HDNη satisfies rrHDNη≤η−1η2−η−1+2η−3η−2+η−22+2.Proof.
First, let us partition the face vertices ofHDNη into four disjoint sets V1=xσℷ/σ=1,2,…,η−1,ℷ=1,2,…,2η−1, V2=xσ2η−1+ℷ/σ=1,2,…,η−1,ℷ=1,2,…,2η−1, V3=xη+σ−1ℷ+1/σ=1,2,…,ℷ/2,ℷ=1,2,…,2η−3, and V4=xη+σ−1ℷ+2η−1/σ=1,2,…,η−ℷ+1/2,ℷ=1,2,…,2η−3. The remaining vertices in HDNη are partitioned into 5 disjoint sets as in Theorem 1.
Now, we define a mappingℸ:VHDNη⟶N∪0 as follows:(3)ℸxσℷ=η−1η2−η−1+η−2σ−1+η−1ℷ−1+2,σ=1,2,…,η−1,ℷ=1,2,…,2η−1,ℸxσ2η−1+ℷ=η−1η2−η−1+η−2σ−1+η−1ℷ−1+2,σ=1,2,…,η−1,ℷ=1,2,…,2η−1,ℸxη+σ−1ℷ+1=η−1η2−η−1+η−2σ−1+η−1ℷ−1+2,σ=1,2,…,ℷ2,ℷ=1,2,…,2η−3,ℸxσ2η−1+ℷ=η−1η2−η−1+η−2σ−1+η−1ℷ−1+2,σ=1,2,…,η−ℷ+12,ℷ=1,2,…,2η−3.
The remaining vertices are labelled as in Theorem1. See Figure 3(b).
Next, we claim thatℸx−ℸy+dx,y≥η∀x,y∈VHDNη.
Letx,y∈VHDNη.Case 1: ifx,y∈V1, then x=xrs and y=xpq, where 1≤r,p≤η−1, 1≤s,q≤2η−1. Therefore, ℸx=η−1η2−η−1+η−2r−1+η−1s−1+2, ℸy=η−1η2−η−1+η−2p−1+η−1q−1+2, and dx,y≥2. If r≠p and s=q, then(4)ℸx−ℸy+dx,y,≥η−1η2−η−1+η−2r−1+η−1s−1+2−η−1η2−η−1,+η−2p−1+η−1q−1+2+2,≥−2+ηr−p+2,≥η.Ifs≠q and r=p, then ℸx−ℸy+dx,y≥η−2η−1s−q+2>η. If both r≠p and s≠q, the condition is obviously true.Case 2: ifx,y∈V2, then x=xr2η−1+s and y=xp2η−1+q, where 1≤r,p≤η−1, 1≤s,q≤2η−1. Here, dx,y≥2. Furthermore, ℸx−ℸy=η−1η2−η−1+η−2r−1+η−1s−1+2−η−1η2−η−1+η−2p−1+η−1q−1+2. The verification for different possibilities is the same as in Case 1.Case 3: supposex,y∈V3; then, dx,y≥2,ℸx=η−1η2−η−1+η−2r−1+η−1s−1+2, and ℸy=η−1η2−η−1+η−2p−1+η−1q−1+2, where 1≤r,p≤2η−3/2, 1≤s,q≤2η−1. If either r≠p or s≠q, then ℸx−ℸy+dx,y≥η−2r−p+2≥η or ℸx−ℸy+dx,y≥η−2ηs−q+2>η.Case 4: ifx,y∈V4, thenℸx=ℸxη+r−12η−1+s=η−1η2−η−1+η−2r−1+η−1s−1+2 and ℸy=ℸxη+p−12η−1+q=η−1η2−η−1+η−2p−1+η−1q−1+2, where 1≤r,p≤η−2η−3+1/2,1≤s,q≤2η−3. Moreover, dx,y≥2. Hence, as in Case 3, we can verify that ℸxℸy+dx,y≥η.Case 5: ifx∈V1 and y∈V2, then x=xrs and y=xpq+2η−1, where 1≤r,p≤η−1, 1≤s,q≤2η−1. Therefore, ℸx=η−1η2−η−1+η−2r−1+η−1s−1+2 and ℸy=η−1η2−η−1+η−2p−1+η−1q−1+2, respectively. Since dx,y≥η and ℸx−ℸy≥0, we get ℸx−ℸy+dx,y≥0+η≥η.Case 6: supposex∈V1 and y∈V3; then,(5)ℸxrs=1+ηη2−η−1+η−2r−1+η−1s−1+2,ℸxη+p−1q+1=η−1η2−η−1+η−2p−1+η−1q−1+2,where1≤r≤−1, 1≤s≤2η−1, 1≤p≤2η−3/2,and1≤q≤2η−3. If r=pands=q, then dx,y=η. Again, if either r≠p or s≠q, then ℸxℸy≥η−2 and dx,y≥2. From the above possibilities, ℸx−ℸy+dx,y≥η.Case 7: ifx∈V1 and y∈V4, then ℸxrs=η−1η2−η−1+η−2r−1+η−1s−1+2 and ℸxη+p−12η−1+q=η−1η2−η−1+η−2p−1+η−1q−1+2, where 1≤r≤η+1, 1≤s≤2η−1, 1≤p≤η−2η−3+1/2,and1≤q≤2η−3. Since dx,y≥η, there is nothing to verify.Case 8: ifx∈V2 and y∈V3, then x=xr2η−1+s and y=xη+p−1q+1, where 1≤r≤η−1, 1≤s≤2η−1, 1≤p≤2η−3/2,and1≤q≤2η−3. Therefore, ℸx=η−1+η−1ηs−1+1,ℸy=η−1η2−η−1+η−2p−1+η−1q−1+2, and dx,y≥η. Hence, ℸx−ℸy+dx,y≥η.Case 9: ifx∈V2 and y∈V4, then the verification is similar to Case 6.Case 10: ifx∈V3 and y∈V4, then ℸx−ℸy=η−1r−1+η−1ηs−1+1−η−1η2−η−1+η−2p−1+−1+ηq−1+2, wherex=xη+r−12η−1+sandy=xη+p−12η−1+q,1≤r≤2η−3/2,1≤s≤2η−3,1≤p≤η−2η−3+1/2,and1≤q≤2η−3. If s=p, then dx,y=η. Therefore, ℸxℸy+dx,y≥η−1r−1+1−η−1q−1+η−1≥η. If s≠p, then dx,y≥2 and ℸxℸy≥η−1η−2 which implies ℸx−ℸy+dx,y≥η.Case 11: letx∈Vs, 1≤s≤4, and y∈Wℷ,1≤ℷ≤5; then, ℸxℸy≥η−1η2−η−1+1>η.Case 12: ifx,y∈Wℷ,1≤ℷ≤5, then the proof is the same as in Theorem 1.
Thus,ℸx−ℸy+dx,y≥η∀x,y∈VHDNη. Hence, ℸ is a valid radial radio labelling, and all the three vertices xη−12η−1,x2η−12η−1,andxη−14η−1 yield the maximum value, η−1η2−η−1+η+2η−1−1+η−12η−1−1+2=−1+ηη2−η−1+2η−3−2+η+η−22+2. Therefore, rrHDNη≤η−1η2−η−1+2η−3η−2+η−22+2.Figure 3
A radial radio labelling ofHDN4 and EDH4 which attains the bound.
(a)(b)Theorem 3.
The radial radio number ofEDHη satisfies rrEDHη≤2η−3ηη−2+2η−12+1,η>1.Proof.
Define a mappingℸ:VEDHη⟶N∪0 as follows:(6)ℸyσℷ=2η−3σ−1+2η−3η−1ℷ−1,σ=1,2,…,η−1,ℷ=1,2,…,η−1,ℸyση+ℷ=2η−3σ−1+2η−3η−1ℷ+2,σ=1,2,…,η−1,ℷ=1,2,…,η−2,ℸyη+σ−1ℷ=2η−3σ−1+2η−3ℷ−1η−1+1,σ=1,2,…,ℷ,ℷ=1,2,…,η−2,ℸyη+σ−1η+ℷ=2η−3σ−1+2η−3ℷ−1η−1+3,σ=1,2,…,η−ℷ,ℷ=1,2,…,η−2,ℸyℷ−1η−1+ση=2η−3σ−1+2,σ=1,2,…,η−1,ℷ=1,2,ℸyσ2η−1=2η−3σ−1,σ=1,2,…,η−1,ℸy2η−1η=2,yη2η−1=1,ℸxσℷ=ηη−22η−3+2η−3σ+2η−3η−1ℷ−1,σ=1,2,…,η−1,ℷ=1,2,…,2η−1,ℸxσ2η−1+ℷ=ηη−22η−3+2η−3σ+2η−3η−1ℷ−1+1,σ=1,2,…,η−1,ℷ=1,2,…,2η−1,ℸxη+σ−1ℷ+1=ηη−22η−3+2η−3σ+2η−3η−1ℷ+1,σ=1,2,…,ℷ2,ℷ=1,2,…,2η−3,ℸxσ2η−1+ℷ=ηη−22η−3+2η−3σ+2η−3η−1ℷ−1+2,σ=1,2,…,η−ℷ+12,ℷ=1,2,…,2η−3.
See Figure2(b).
As the remaining part of the proof is similar to Theorem2, we have left the proof to the reader.


## 4. Conclusion

In this research study, we have introduced the enhanced hexagonal difference hexagonal network from the existing enhanced hexagonal network. Furthermore, we have investigatedrrEDHη≤2η−3ηη−2+2η−12+1,η>1. Also, the radial radio number of HDNη and EDHη was presented. These bounds will motivate other researchers to conduct further research studies on the applications of the enhanced hexagonal network and its derived networks. Furthermore, these studies can be extended to the radio k-chromatic number and its variations.



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