# Radio and Radial Radio Numbers of Certain Sunflower Extended Graphs

Authors: Mohammed K. A. Kaabar; Kins Yenoke
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

Communication systems including AM and FM radio stations transmitting signals are capable of generating interference due to unwanted radio frequency signals. To avoid such interferences and maximize the number of channels for a predefined spectrum bandwidth, the radio-k-chromatic number problem is introduced. Let G=V,E be a connected graph with diameter d and radius ρ. For any integer k, 1≤k≤d, radio k−coloring of G is an assignment φ of color (positive integer) to the vertices of G such that da,b+φa−φb≥1+k, ∀a,b∈VG, where da,b is the distance between a and b in G. The biggest natural number in the range of φ is called the radio k−chromatic number of G, and it is symbolized by rckφ. The minimum number is taken over all such radio k−chromatic numbers of φ which is called the radio k−chromatic number, denoted by rckG. For k=d and k=ρ, the radio k−chromatic numbers are termed as the radio number (rnG) and radial radio number (rrG) of G, respectively. In this research work, the relationship between the radio number and radial radio number is studied for any connected graph. Then, several sunflower extended graphs are defined, and the upper bounds of the radio number and radial radio number are investigated for these graphs.


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## Body

## 1. Introduction

The channel frequency assignment problem was first proposed by Griggs and Yeh [1] in 1992 for the amplitude modulation radio stations. Due to the cochannel interference, there is a challenge to fix the transmitters in a particular geographical area. Therefore, studying the channel assignment problem in radio stations is NP‐complete. However, Fotakis et al. [2] proved that even for graphs with diameter 2, the problem is NP-hard. Chartrand et al. [3] presented the theoretical graph definition for the radio-k-chromatic number as follows.LetG=V,E be a connected graph with diameter d and radius ρ. For any integer k, 1≤k≤d, radio k−coloring of G is an assignment φ of color (positive integer) to the vertices of G such that da,b+φa−φb≥1+k, ∀a,b∈VG, where da,b is the distance between a and b in G. The biggest natural number in the range of φ is called the radio k−chromatic number of G, and it is symbolized by rckφ. The minimum number is taken over all such radio k−chromatic numbers of φ which is called the radio k−chromatic number, denoted by rckG.Čada et al. [4] proved that, for any distance graph Dt−1,t, we have(1)rckDt−1,t≤12k2+k−t+22,wheret>2is an integer,k>3is an odd integer,12tk2,k>0is an odd integer,t>3is an even integer.Recently, Bantva [5] improved this general lower bound. Based on different k values, the radio k−chromatic number is classified into different problems.Fork=d, the radio k−chromatic number is termed as the radio number problem, and it is symbolized by rnG. It was introduced by Chartrand et al. [6] for the purpose of determining the maximum number of channels for frequency modulation (FM) radio stations by minimum utilization of spectrum bandwidth. The radio number problem has been studied by several researchers [7, 8]. In 2017, Avadayappan et al. [9] brought in the concept of radial radio labelling. A mapping φ:VG⟶N∪0 for a connected graph G=V,E is called a radial radio labelling if this satisfies the inequality φa−φb+da,b≥ρ+1∀a,b∈VG, where ρ is the radius of the graph G. Radial radio number of φ symbolized by rrφ is the maximum number mapped under φ. The radial radio number of G, denoted by rrG, is equal to minrrφ/φis a radial radio labelling ofG. A few number of research articles [10, 11] were published in the area of radial radio labelling. In this paper, we have studied a comparative relation between rnG and rrG. Furthermore, we have defined and determined the radio and radial radio numbers of certain sunflower extended graphs such as SSn,Ւ,CSn,Ւ, and WSn,Ւ.


## 2. Relation between the Radio Number and Radial Radio Number

This section deals with certain results which connectrnG with rrG for any connected graph G.Definition 1.
The eccentricity of a vertexz, represented by ez in a connected graph G, is the maximum distance from z to any other vertex in G. That is, ez=maxdz,a∀a∈VG. The maximum eccentricity of the vertices of G is called the diameter of the graph, and it is symbolized by d or diamG. In addition, the radius of graph G, symbolized by ρ or radG, is the minimum eccentricity of the vertices of G.Definition 2.
A connected graphG=V,E is called a self-centred graph if eu=ev∀u,v∈VG. In other words, diamG=radG.
The following is a straight result from the definitions of the radio number and radial radio number.Theorem 1.
For any connected graphG,rnG≥rrG.
Chartrand et al. [6] proved the following three theorems, which will be used to study the general results for the radial radio number.Theorem 2.
IfG is a connected graph of order n and diameter d, then n≤rnG≤n−1d.Theorem 3.
For a completek−partite graph G of order n, rnG=n+k+1.Theorem 4.
Every connected graphG of order n with rnG=n is self-centred.Using Theorem5 and Definition 2, we have attained the equality of Theorem 1 as follows.Theorem 5.
A connected graphG of order n is self-centred if and only if rnG=rrG=n.Theorem 6.
LetG=V,E be a complete k−partite graph of order n; then, rrG=k.Proof.
Let the vertex set ofG be partitioned into k disjoint sets U1,U2,…,Uk such that Ui∩Uj=∅,1≤i≠j≤k, and V=∪i=1kUi. The radius of the complete k−partite graph is 1, and all the vertices in the sets Ui,1≤i≤k, are at distance two. Hence, we can label the vertices in each set Ui as ii=1,2,…,k. Clearly, the radial radio labelling condition da,b+φa−φb≥2 is satisfied for any pair of vertices in G. Hence, rrG=k.Theorem 7.
IfG is a connected graph of order n>1 and radius ρ, then 2≤rrG≤n−1ρ.Proof.
GivenG is a connected graph that contains at least two vertices. Therefore, the lower bound of the theorem attains in the particular case of Theorem 6 which is for the complete bipartite graphs. Furthermore, the upper bound is obtained by replacing d by ρ in Theorem 2. Consequently, 2≤rrG≤n−1ρ,n>1.


## 3. Results and Discussion

In this section, we have defined and investigated the radial radio and radio number of some sunflower extended graphs such as star-sun graphSSn,Ւ, complete-sun graph CSn,Ւ, wheel-sun graph WSn,Ւ, and fan-sun graph FSn,Ւ.Definition 3.
A sunflower graph consists of a wheel with a centre vertexwn, n-cycle w0,w1,…,wn−1, and additional n vertices u0,u1,…,un−1 where ui is joined with edges to (wi,wi+1), i=0,1,2,…n–1, and i+1 is taken as modulo n. It is represented by SFn. The radius, diameter, and number of vertices of SFn are 2, 4, and 2n+1, respectively.Definition 4.
A star graph, denoted by SՒ+1, is defined as a complete bipartite graph of the form K1,Ւ,Ւ>1. In other words, SՒ+1 is a tree having Ւ leaves and one internal vertex.Definition 5.
A star-sun graph, denoted bySSn,Ւ, is a graph obtained from the sunflower graph SFn and n copies of star graph SՒ+1 by merging the internal vertex of the kth star graph SՒ+1 and vertex uk−1 of SFn, 1≤k≤n, as shown in Figure 1(a).Figure 1
Different sunflower extended graphs.
(a)(b)(c)(d)Remark 1.
The cardinality ofVSSn,Ւ and ESSn,Ւ in SSn,Ւ is nՒ+2+1 and 2nՒ+4n, respectively. Also, the diameter and radius of the graph are 6 and 3, respectively.Definition 6.
A complete-sun graph, denoted byCSn,Ւ, is a graph obtained from the sunflower graph SFn and n copies of complete graph KՒ by merging a vertex of the mth complete graph KՒ and the vertex um−1 of SFn, 1≤m≤n, as shown in Figure 1(b). Here, we have VCSn,Ւ=nՒ+1 and ECSn,Ւ=nՒՒ−1/2+2n.Remark 2.
The diameter and radius ofCSn,Ւ are 6 and 3, respectively.Definition 7.
A wheel-sun graph, denoted byWSn,Ւ, is a graph obtained from the sunflower graph SFn and n copies of wheel graph WՒ+1 by merging the vertex uk−1 of SFn and the centre vertex of the kth wheel, where 1≤k≤n as shown in Figure 1(c).Remark 3.
The number of vertices inWSn,Ւ is nՒ+2+1, while its number of edges is 2n2n+Ւ. Also, its diameter and radius are 6 and 3, respectively.Definition 8.
A fan-sun graph is a graph obtained from the sunflower graphSFn and n copies of fan graph FՒ+1=PՒ+K1 by merging K1 of the kth fan and the vertex uk−1 of SFn, 1≤k≤n. It is denoted by FSn,Ւ as shown in Figure 1(d).Remark 4.
For the graphFSn,Ւ, the number of edges is 4n+n2Ւ−1, while the number of vertices is nՒ+2+1. Moreover, the diameter and radius are 6 and 3, respectively.
In this work, we name the newly includednՒ vertices of SSn,Ւ, WSn,Ւ, and FSn,Ւ as v1,v2,…,vnՒ in the clockwise sense.


### 3.1. Radial Radio Number of Sunflower Extended Graphs

The following theorems provide the upper bound for the radial radio number ofSn,Ւ, CSn,Ւ, andWSn,Ւ.Theorem 8.
LetG be the sun-star graph SSn,Ւ. Then, rrSSn,Ւ≤3n+2Ւ+1.Proof.
First, we define a mappingφ:VSSn,Ւ⟶N as follows: φvՒj−1+i=2i,i=1,2,…,Ւ,j=1,2,…,n, φw2i=2Ւ+1+i,i=0,1,…,⌈n/2⌉−1, φw2i+1=2⌈n/2⌉+Ւ+1+i,i=0,1,…,⌊n/2⌋−1, φu2i=2n+Ւ+1+i,i=0,1,…,⌈n/2⌉−1, φu2i+1=2n+Ւ+1+⌈n/2⌉+i,i=0,1,2,…,⌊n/2⌋−1, and φwn=1 as shown in Figure 2.
Since the radius of the graph is 3, we must verifyφ satisfies the radial radio labelling condition da,b+φa−φb≥1+radSSn,Ւ=1+3=4 for every pair of vertices a,b∈VSSn,Ւ.
Let us choose any two arbitrary verticesa and b in the sun-star graph.Case 1: supposea and b are star vertices, then a and b are of the form a=vl and b=vm,1≤m≠l≤nՒ.Case 1.1: ifl=Ւj−1+αandm=Ւj−1+β,α≠β, then the value of a and b under φ is 2p and 2q, respectively. Also, a and b are at a distance two. Hence, the radial radio labelling condition becomes da,b+φa−φb=2+2α−β≥4 since α≠β.Case 1.2: ifl=α+Ւs−1andm=β+Ւt−1,t≠s, then a and b are at a distance at least 4. Hence, the radial radio labelling condition is trivially satisfied.Case 2: leta=vj−1+l, 1≤l≤Ւ, and b=wm,0≤m≤n−1; then, the value of φa is 2l, and φb is at least 2Ւ+1+m. Furthermore, da,b is at least 2. Therefore, da,b+φa−φb≥2+2Ւ−2Ւ+2≥4.Case 3: if we takea=vՒj−1+s, 1≤s≤Ւ, and b=ut,0≤t≤n−1, then φa−φb≥2n+Ւ+1−2Ւ≥2n>4 since n>2, which trivially verifies the radial radio labelling condition.Case 4: supposea=vՒj−1+l, 1≤l≤Ւ, and b=wm,0≤m≤n−1, then the value of φa is 2l, and φb is at least 2Ւ+1+m. Furthermore, da,b is at least 2. Therefore, da,b+φa−φb≥2+2Ւ+2−2Ւ≥4.Case 5: ifa is the centre vertex of the wheel and b is any other star vertex, then the distance between them is exactly 3. Also, φa=0 and φb≥1. Therefore, da,b+φa−φb≥4.Case 6: leta and b be the vertices in the n-cycle of the sunflower graph.Case 6.1: ifa=w2l and b=w2m, 0≤l≠m≤⌈n/2⌉−1, then φa−φb≥2l−m. Again, da,b=2. Since l≠m, the condition for the radial radio labelling is satisfied.Case 6.2: supposea and b are of the form w2l+1 and w2m+1, where 0≤l≠m≤⌊n/2⌋−1, then the distance between them is exactly two. Also, the function values of a and b are 2Ւ+l+⌈n/2⌉+1 and 2Ւ+m+⌈n/2⌉+1, respectively. Hence, the radial radio labelling condition becomes da,b+φa−φb=2+2⌈n/2⌉+Ւ+l+1−2⌈n/2⌉+Ւ+m+1≥4.Case 6.3: supposea=w2l, 0≤l≤⌈n/2⌉−1, and b=w2m+1, 0≤m≤⌊n/2⌋−1, then φa=2Ւ+1+l and φb=2Ւ+m+⌈n/2⌉+1.
Ifm=0 and l=⌈n/2⌉−1, then da,b=2; else, da,b≥1. Hence, in both possibilities, we obtain da,b+φa−φb=2+2⌈n/2⌉+Ւ+1−2Ւ+⌈n/2⌉−1+1≥2+2=4 and da,b+φa−φb≥1+2⌈n/2⌉+Ւ+m−2Ւ+l+1≥1+4>4.Case 7: ifa=wl,0≤l≤n−1, and b=wn, then φa−φb is at least 2Ւ+1. Since n>2, the radial radio labelling condition is easily verified.Case 8: supposea=ul and =um,0≤l≠m≤n−1.Case 8.1: ifl=2α and m=2β, 0≤α≠β≤⌈n/2⌉−1, respectively, then φa=2n+Ւ+1+α and φb=2n+Ւ+1+β. Also, the distance between a and b is 2. Hence, we have da,b+φa−φb=2+2n+Ւ+1+α−2n+Ւ+1+β>3 since α≠β.Case 8.2: ifl=2α+1 and m=2β+1, 0≤α≠β≤⌈n/2⌉−1, then da,b=2. Also, φ takes the values of a and b to 2n+Ւ+⌈n/2⌉+α+1 and 2n+Ւ+1+⌈n/2⌉+β+1, respectively. Therefore, da,b+φa−φb=2+2n+Ւ+1+⌈n/2⌉+α+1−2n+Ւ+1+⌈n/2⌉+β+1≥4.Case 8.3: leta=u2l, 0≤l≤⌈n/2⌉−1, and b=u2m+1, 0≤m≤⌊n/2⌋−1; then, a and b are mapped to 2n+Ւ+1+l and2n+Ւ+1+⌈n/2⌉+m+1, respectively.Whenl=⌈n/2⌉−1 and m=0, da,b=2, and hence, da,b+φa−φb=2+2n+Ւ+1+⌈n/2⌉−1−2n+Ւ+1+⌈n/2⌉+1≥2+2=4.The remaining possibility is obvious since the distance between them is at least 1, and the modulus difference betweenφa and φb is at least 4.Case 9: ifa=wn and b is any vertex of the form ul, 0≤m≤n−1, then the verification is obvious since the difference in the function values of a and b is at least 2n+Ւ.Thus,φ is a valid radial radio labelling, and the vertex un−1 receives the maximum number 2n+Ւ+1+⌈n/2⌉+⌊n/2⌋−1=3n+2Ւ−1. That is, rrφ=3n+2Ւ+1.
Hence, we conclude thatrrSSn,Ւ≤3n+2Ւ+1.Figure 2
A sun-star graphSS6,5 and its radial radio labelling.
(a)(b)Theorem 9.
LetG be the complete-sun graph CSn,Ւ. If n≡0mod3, then the radial radio number of G satisfies rrG≤3Ւ+1+2n.Proof.
Let us name the newly includednՒ−1 vertices of CSn,Ւ as v1,v2…vnՒ−1 in the clockwise sense. Now, we define a one-one mapping φ:VCSn,Ւ⟶1,2… as follows:(2)φvՒ−1j−1+i=3i−1,i=1,2,…,Ւ−1,j=1,2…n,φu3i+2=3Ւ−1,i=0,1,…,n3−1,φu3i=3Ւ+1,i=0,1,…,n3−1,φu3i+1=3Ւ+3,i=0,1,…,n3−1,φw2i=3Ւ+2i+5,i=0,1,…,n2−1,φw2i+1=3Ւ+2n2+2i+5,i=0,1,…,n2−1,φwn=1.
This mapping is visible in Figure3(a).
In the following, we claim thatda,b+φa−φb≥4∀a,b∈VCSn,Ւ.
Leta,b∈VCSn,Ւ.Case 1: supposea=vՒ−1p−1+α and b=vՒ−1q−1+β,1≤α,β≤Ւ−1, 1≤p,q≤n, then φvՒ−1p−1+l=3l−1 and φvՒ−1q−1+m=3m−1.Case 1.1: ifα≠β, then dvՒ−1p−1+α,vՒ−1q−1+β=1 and φa−φb≥3α−β. Therefore, da,b+φa−φb≥1+3α−β≥4 since α≠β.Case 1.2: ifp=q, then dvՒ−1p−1+α,vՒ−1q−1+β=4, which is enough for verifying the condition.Case 2: assume thata=vՒ−1p−1+α, 1≤α≤Ւ−1, 1≤p≤n, and b=uβ,0≤β≤n−1; then, φa≤3Ւ−4 and φb≥3Ւ−1. Also, da,b≥1. Therefore, da,b+φa−φb≥1+3≥4.Case 3: if we take=vՒ−1p−1+α, 1≤α≤Ւ−1, 1≤p≤n, and b=wβ,0≤β≤n−1, then φa−φb≥3Ւ−4−3Ւ+3>4 since n>2, which verifies the condition trivially.Case 4: assume thata=uα and b=wβ,0≤α,β≤n−1.Case 4.1: ifα=n−2 and β=1, then φuα=3Ւ+3 and φwβ=3Ւ+5. However, duα,wβ=2. Therefore, da,b+φa−φb≥4. Otherwise, φa−φb is greater than 3.Case 5: leta=uα and b=uβ, 0≤α≠β≤n−1. In this case, if α=3p+2 and β=3q+2 or α=3p+1 and β=3q+1 or α=3p and β=3q, 0 ≤p≠q≤n/3−1, then da,b=4 and φa−φb=0. Otherwise, da,b≥2 and φa−φb≥2. Hence, in both possibilities, the condition for radial radio labelling is satisfied.Case 6: assume thata=wα and b=wβ, 0≤α≠β≤n−1. If α=2p and β=2q or α=2p+1 and β=2q+1, 0≤p≠q, 0≤p≠q≤⌈n/2⌉−1, then da,b=2. Thus, da,b=2 and φa−φb≥2. Otherwise, da,b≥1 and φa−φb≥4. Therefore, the condition holds in both of the possibilities.Case 7: finally, let us assume thata=wn, and b is any vertex in CSn,Ւ. If b=v1, then da,b=3 and φa−φb=1. Otherwise, the radial radio labelling condition is obviously true. Thus, φ satisfies the condition of radial radio labelling and attains the maximum value 3Ւ+2⌈n/2⌉+2⌊n/2⌋−2+5=3Ւ+2n+3 for the vertex w2⌊n/2⌋. Therefore, we get rrCSn,Ւ≤3Ւ+1+2n.
The proof for the other two cases, namely,n≡1mod3 and n≡2mod3, is left to the reader.Figure 3
A radial radio labelling of complete-sun graphCSn,Ւ for n=9andՒ=4 and wheel-sun graph WS8,5.
(a)(b)Theorem 10.
Forn>2andr>3, the radial radio number of the wheel-star graph satisfies(3)rrSSn,Ւ≤2Ւ+n+6,n≡0mod32Ւ+n+8,n≡1mod32Ւ+n+9,n≡2mod3.
We omit the proof, but Figure3(b) illustrates the case n≡2mod3.


### 3.2. Radio Number of Sunflower Extended Graphs

This section provides the upper bound for the radio number ofSn,Ւ, SSn,Ւ,andWSn,Ւ.Theorem 11.
Forn>3 and n≡0mod3, the radio number of the complete-sun graph satisfies rrCSn,Ւ≤18Ւ+9n−30.Proof.
We define a 1-1 mappingφ:VCSn,Ւ⟶1,2… as follows:(4)φv3Ւ−1j−1+i=6i−1+j+4,i=1,2,…,Ւ−1,j=1,2,…n3,φvՒ−13j−2+i=6Ւ+n3+6i−1+j−8,i=1,2,…,Ւ−1,j=1,2,…,n3,φvՒ−13j−1+i=12Ւ+2n3+6i−1+j−20,i=1,2,…,Ւ−1,j=1,2,…,n3,φu3i=18Ւ+n+3i−28,i=0,1,…,n3−1,φu3i+1=18Ւ+2n+3i−28,i=0,1,…,n3−1,φu3i+2=18Ւ+3n+i−28,i=0,1,…,n3−1,φw2i=18Ւ+4n+5i−25,i=0,1,…,n2−1,φw2i+1=18Ւ+4n+5n2+5i−25,i=0,1,…,n2−1,φwn=1.
See Figure3(a).
Then, to showφ is a valid radio labelling, we must verify the inequality(5)da,b+φa−φb≥7∀a,b∈VCSn,Ւ.
Leta,b∈VCSn,Ւ.Case 1: suppose thata=vα and b=vβ,1≤α≠β≤nՒ−1.Case 1.1: ifα=3Ւ−1s−1+pandβ=3Ւ−1t−1+q or α=Ւ−13s−2+pandβ=Ւ−13s−2+p or α=Ւ−13s−1+pandβ=Ւ−13s−1+p where p≠q and s=t, then dvα,vβ=1 and φvα−φvβ≥6p−q≥6. In the same subcase, if s≠t, then dvα,vβ=6 and φvα−φvβ≥1. So, da,b+φa−φb≥7.Case 1.2: ifα=3Ւ−1s−1+pandβ=3Ւ−1t−1+q,where1≤p,q≤Ւ−1, 1≤s,t≤n/3, then φvα=6p−1+s+4andφvβ=6Ւ+n/3+6q−1+t−8. In addition, dvα,vβ≥4. Therefore, da,b+φa−φb≥4+6p−1+s+4−6Ւ+n/3+6q−1+t−8≥4+3=7.Case 1.3: ifα=3Ւ−1s−1+pandβ=Ւ−13t−1+q,where1≤p,q≤Ւ−1, 1≤s,t≤n/3, then φa−φb=6p−1+s+4−12Ւ+2n/3+6q−1+t−20>6. Consequently, the condition is true.Case 1.4: ifα=Ւ−13s−2+pandβ=Ւ−13t−1+q,where1≤p,q≤Ւ−1, 1≤s,t≤, then dvα,vβ≥4 and φvα−φvβ=6Ւ+n/3+6p−1+s−8−12Ւ+2n/3+6q−1+t−20≥3. It follows that da,b+φa−φb≥7.Case 2: takea=vα and b=vβ,0≤α,β≤n−1. If α=3s+pandβ=3t+q,0≤p=q≤2, 0≤s,t≤n/3−1, then dv3s+p,v3t+p=4 and φv3s+p−φv3t+p≥3. Otherwise, that is, if p≠q, thus, dv3s+p,v3t+q≥2 and φv3s+p−φv3t+q≥n>6sincen>6. Therefore, in both chances, we get da,b+φa−φb≥7.Case 3: assume thata=u2s+p and b=u2t+q,0≤s≤⌈n/2⌉−1,0≤t≤⌊n/2⌋−1,0≤p,q≤1.Ifp=q, then φu2s+p−φu2t+q≥4 and dv3s+p,v3t+p=3; else, dv3s+p,v3t+q≥2 and φv3s+p−φv3t+q≥⌈n/2⌉+⌊n/2⌋. Therefore, in both of them, the inequality is satisfied.Case 4: suppose thata=vα and b=uβ,0≤α≤nՒ−1, 0≤β≤n−1; then, either da,b=3 and φa−φb≥4 or da,b=1 and φa−φb>6. Therefore, da,b+φa−φb≥7.Case 5: if we takea=vα, 1≤α≤nՒ−1, and b=wβ, 0≤β≤n−1, then from the mapping, φa−φb≥12Ւ+2n/3+6Ւ−2+n/3−20−18Ւ+4n−25φ>n>6, which verifies the condition trivially.Case 6: assume thata=uα and b=wβ, 0≤α,β≤n−1. If α=n−1 and β=0, then φuα−φwβ=18Ւ+3n+n/3−1−28−18Ւ+4n+5⌈n/2⌉−25=6 and duα,wβ=1. Otherwise, φuα−φwβ>6. So, duα,wβ+φuα−φwβ>6.Case 7: leta be the centre vertex of the wheel and b be any other vertex in the graph. If b=v1, then φwn−φv1=1−5=4andda,b=3. Otherwise, the condition is obviously true. Thus, φ is a valid radio labelling, and the vertex wn−1 is labelled with the maximum number 18Ւ+4n+5⌊n/2⌋+5⌈n/2⌉−1−25=18Ւ+9n−30. Hence, rnCSn,Ւ≤18Ւ+9n−30, n>4 and n≡0mod3.Theorem 12.
Forn>5 and n≡1mod3, the radio number of the star-sun graph satisfies rrSSn,Ւ≤20Ւ+12⌊n/3⌋+5n−4.Proof.
First, we define an injectionφ:VSSn,Ւ⟶N as follows:(6)φv3Ւj−1+i=5i−1+j+4,i=1,2,…,Ւ,j=1,2,…,n3,φvՒ3j−2+i=5Ւ+n3+5i−1+j−1,i=1,2,…Ւ,j=1,2,…,n3,φvՒ−13j−1+i=10Ւ+2n3+5i−1+j−4,i=1,2,…,Ւ,j=1,2,…,n3,φvՒn−1+i=15Ւ+3n3+5i−1−6,i=1,2,…,Ւ,φu3i=20Ւ+3n3+3i−7,i=0,1,…,n3,φu3i+1=20Ւ+6n3+3i−4,i=0,1,…,n3,φu3i+2=20Ւ+9n3+3i−1,i=0,1,…,n3−1,φw2i=20Ւ+12n3+5i+1,i=0,1,…,n2−1,φw2i+1=20Ւ+12n3+5n2+i+1,i=0,1,…,n2−1,φwn=1.
We omit the rest of the proof as it is similar to the one for Theorem11.
The other two cases in Theorems11 and 12 are also left to the reader.Theorem 13.
Forn>5, the radio number of the wheel-sun graph satisfies(7)rrWSn,Ւ≤15Ւ+n3+8n−4,n≡0mod319Ւ+n3+8n−4,n≡1mod3.19+n3+8n−5,n≡2mod3
Figure4 illustrates the case n≡2mod3. We omit the proof of this theorem.Figure 4
A radio labelling of complete-sun graphCS9,4 and wheel-sun graph WS8,5.


## 3.1. Radial Radio Number of Sunflower Extended Graphs

The following theorems provide the upper bound for the radial radio number ofSn,Ւ, CSn,Ւ, andWSn,Ւ.Theorem 8.
LetG be the sun-star graph SSn,Ւ. Then, rrSSn,Ւ≤3n+2Ւ+1.Proof.
First, we define a mappingφ:VSSn,Ւ⟶N as follows: φvՒj−1+i=2i,i=1,2,…,Ւ,j=1,2,…,n, φw2i=2Ւ+1+i,i=0,1,…,⌈n/2⌉−1, φw2i+1=2⌈n/2⌉+Ւ+1+i,i=0,1,…,⌊n/2⌋−1, φu2i=2n+Ւ+1+i,i=0,1,…,⌈n/2⌉−1, φu2i+1=2n+Ւ+1+⌈n/2⌉+i,i=0,1,2,…,⌊n/2⌋−1, and φwn=1 as shown in Figure 2.
Since the radius of the graph is 3, we must verifyφ satisfies the radial radio labelling condition da,b+φa−φb≥1+radSSn,Ւ=1+3=4 for every pair of vertices a,b∈VSSn,Ւ.
Let us choose any two arbitrary verticesa and b in the sun-star graph.Case 1: supposea and b are star vertices, then a and b are of the form a=vl and b=vm,1≤m≠l≤nՒ.Case 1.1: ifl=Ւj−1+αandm=Ւj−1+β,α≠β, then the value of a and b under φ is 2p and 2q, respectively. Also, a and b are at a distance two. Hence, the radial radio labelling condition becomes da,b+φa−φb=2+2α−β≥4 since α≠β.Case 1.2: ifl=α+Ւs−1andm=β+Ւt−1,t≠s, then a and b are at a distance at least 4. Hence, the radial radio labelling condition is trivially satisfied.Case 2: leta=vj−1+l, 1≤l≤Ւ, and b=wm,0≤m≤n−1; then, the value of φa is 2l, and φb is at least 2Ւ+1+m. Furthermore, da,b is at least 2. Therefore, da,b+φa−φb≥2+2Ւ−2Ւ+2≥4.Case 3: if we takea=vՒj−1+s, 1≤s≤Ւ, and b=ut,0≤t≤n−1, then φa−φb≥2n+Ւ+1−2Ւ≥2n>4 since n>2, which trivially verifies the radial radio labelling condition.Case 4: supposea=vՒj−1+l, 1≤l≤Ւ, and b=wm,0≤m≤n−1, then the value of φa is 2l, and φb is at least 2Ւ+1+m. Furthermore, da,b is at least 2. Therefore, da,b+φa−φb≥2+2Ւ+2−2Ւ≥4.Case 5: ifa is the centre vertex of the wheel and b is any other star vertex, then the distance between them is exactly 3. Also, φa=0 and φb≥1. Therefore, da,b+φa−φb≥4.Case 6: leta and b be the vertices in the n-cycle of the sunflower graph.Case 6.1: ifa=w2l and b=w2m, 0≤l≠m≤⌈n/2⌉−1, then φa−φb≥2l−m. Again, da,b=2. Since l≠m, the condition for the radial radio labelling is satisfied.Case 6.2: supposea and b are of the form w2l+1 and w2m+1, where 0≤l≠m≤⌊n/2⌋−1, then the distance between them is exactly two. Also, the function values of a and b are 2Ւ+l+⌈n/2⌉+1 and 2Ւ+m+⌈n/2⌉+1, respectively. Hence, the radial radio labelling condition becomes da,b+φa−φb=2+2⌈n/2⌉+Ւ+l+1−2⌈n/2⌉+Ւ+m+1≥4.Case 6.3: supposea=w2l, 0≤l≤⌈n/2⌉−1, and b=w2m+1, 0≤m≤⌊n/2⌋−1, then φa=2Ւ+1+l and φb=2Ւ+m+⌈n/2⌉+1.
Ifm=0 and l=⌈n/2⌉−1, then da,b=2; else, da,b≥1. Hence, in both possibilities, we obtain da,b+φa−φb=2+2⌈n/2⌉+Ւ+1−2Ւ+⌈n/2⌉−1+1≥2+2=4 and da,b+φa−φb≥1+2⌈n/2⌉+Ւ+m−2Ւ+l+1≥1+4>4.Case 7: ifa=wl,0≤l≤n−1, and b=wn, then φa−φb is at least 2Ւ+1. Since n>2, the radial radio labelling condition is easily verified.Case 8: supposea=ul and =um,0≤l≠m≤n−1.Case 8.1: ifl=2α and m=2β, 0≤α≠β≤⌈n/2⌉−1, respectively, then φa=2n+Ւ+1+α and φb=2n+Ւ+1+β. Also, the distance between a and b is 2. Hence, we have da,b+φa−φb=2+2n+Ւ+1+α−2n+Ւ+1+β>3 since α≠β.Case 8.2: ifl=2α+1 and m=2β+1, 0≤α≠β≤⌈n/2⌉−1, then da,b=2. Also, φ takes the values of a and b to 2n+Ւ+⌈n/2⌉+α+1 and 2n+Ւ+1+⌈n/2⌉+β+1, respectively. Therefore, da,b+φa−φb=2+2n+Ւ+1+⌈n/2⌉+α+1−2n+Ւ+1+⌈n/2⌉+β+1≥4.Case 8.3: leta=u2l, 0≤l≤⌈n/2⌉−1, and b=u2m+1, 0≤m≤⌊n/2⌋−1; then, a and b are mapped to 2n+Ւ+1+l and2n+Ւ+1+⌈n/2⌉+m+1, respectively.Whenl=⌈n/2⌉−1 and m=0, da,b=2, and hence, da,b+φa−φb=2+2n+Ւ+1+⌈n/2⌉−1−2n+Ւ+1+⌈n/2⌉+1≥2+2=4.The remaining possibility is obvious since the distance between them is at least 1, and the modulus difference betweenφa and φb is at least 4.Case 9: ifa=wn and b is any vertex of the form ul, 0≤m≤n−1, then the verification is obvious since the difference in the function values of a and b is at least 2n+Ւ.Thus,φ is a valid radial radio labelling, and the vertex un−1 receives the maximum number 2n+Ւ+1+⌈n/2⌉+⌊n/2⌋−1=3n+2Ւ−1. That is, rrφ=3n+2Ւ+1.
Hence, we conclude thatrrSSn,Ւ≤3n+2Ւ+1.Figure 2
A sun-star graphSS6,5 and its radial radio labelling.
(a)(b)Theorem 9.
LetG be the complete-sun graph CSn,Ւ. If n≡0mod3, then the radial radio number of G satisfies rrG≤3Ւ+1+2n.Proof.
Let us name the newly includednՒ−1 vertices of CSn,Ւ as v1,v2…vnՒ−1 in the clockwise sense. Now, we define a one-one mapping φ:VCSn,Ւ⟶1,2… as follows:(2)φvՒ−1j−1+i=3i−1,i=1,2,…,Ւ−1,j=1,2…n,φu3i+2=3Ւ−1,i=0,1,…,n3−1,φu3i=3Ւ+1,i=0,1,…,n3−1,φu3i+1=3Ւ+3,i=0,1,…,n3−1,φw2i=3Ւ+2i+5,i=0,1,…,n2−1,φw2i+1=3Ւ+2n2+2i+5,i=0,1,…,n2−1,φwn=1.
This mapping is visible in Figure3(a).
In the following, we claim thatda,b+φa−φb≥4∀a,b∈VCSn,Ւ.
Leta,b∈VCSn,Ւ.Case 1: supposea=vՒ−1p−1+α and b=vՒ−1q−1+β,1≤α,β≤Ւ−1, 1≤p,q≤n, then φvՒ−1p−1+l=3l−1 and φvՒ−1q−1+m=3m−1.Case 1.1: ifα≠β, then dvՒ−1p−1+α,vՒ−1q−1+β=1 and φa−φb≥3α−β. Therefore, da,b+φa−φb≥1+3α−β≥4 since α≠β.Case 1.2: ifp=q, then dvՒ−1p−1+α,vՒ−1q−1+β=4, which is enough for verifying the condition.Case 2: assume thata=vՒ−1p−1+α, 1≤α≤Ւ−1, 1≤p≤n, and b=uβ,0≤β≤n−1; then, φa≤3Ւ−4 and φb≥3Ւ−1. Also, da,b≥1. Therefore, da,b+φa−φb≥1+3≥4.Case 3: if we take=vՒ−1p−1+α, 1≤α≤Ւ−1, 1≤p≤n, and b=wβ,0≤β≤n−1, then φa−φb≥3Ւ−4−3Ւ+3>4 since n>2, which verifies the condition trivially.Case 4: assume thata=uα and b=wβ,0≤α,β≤n−1.Case 4.1: ifα=n−2 and β=1, then φuα=3Ւ+3 and φwβ=3Ւ+5. However, duα,wβ=2. Therefore, da,b+φa−φb≥4. Otherwise, φa−φb is greater than 3.Case 5: leta=uα and b=uβ, 0≤α≠β≤n−1. In this case, if α=3p+2 and β=3q+2 or α=3p+1 and β=3q+1 or α=3p and β=3q, 0 ≤p≠q≤n/3−1, then da,b=4 and φa−φb=0. Otherwise, da,b≥2 and φa−φb≥2. Hence, in both possibilities, the condition for radial radio labelling is satisfied.Case 6: assume thata=wα and b=wβ, 0≤α≠β≤n−1. If α=2p and β=2q or α=2p+1 and β=2q+1, 0≤p≠q, 0≤p≠q≤⌈n/2⌉−1, then da,b=2. Thus, da,b=2 and φa−φb≥2. Otherwise, da,b≥1 and φa−φb≥4. Therefore, the condition holds in both of the possibilities.Case 7: finally, let us assume thata=wn, and b is any vertex in CSn,Ւ. If b=v1, then da,b=3 and φa−φb=1. Otherwise, the radial radio labelling condition is obviously true. Thus, φ satisfies the condition of radial radio labelling and attains the maximum value 3Ւ+2⌈n/2⌉+2⌊n/2⌋−2+5=3Ւ+2n+3 for the vertex w2⌊n/2⌋. Therefore, we get rrCSn,Ւ≤3Ւ+1+2n.
The proof for the other two cases, namely,n≡1mod3 and n≡2mod3, is left to the reader.Figure 3
A radial radio labelling of complete-sun graphCSn,Ւ for n=9andՒ=4 and wheel-sun graph WS8,5.
(a)(b)Theorem 10.
Forn>2andr>3, the radial radio number of the wheel-star graph satisfies(3)rrSSn,Ւ≤2Ւ+n+6,n≡0mod32Ւ+n+8,n≡1mod32Ւ+n+9,n≡2mod3.
We omit the proof, but Figure3(b) illustrates the case n≡2mod3.


## 3.2. Radio Number of Sunflower Extended Graphs

This section provides the upper bound for the radio number ofSn,Ւ, SSn,Ւ,andWSn,Ւ.Theorem 11.
Forn>3 and n≡0mod3, the radio number of the complete-sun graph satisfies rrCSn,Ւ≤18Ւ+9n−30.Proof.
We define a 1-1 mappingφ:VCSn,Ւ⟶1,2… as follows:(4)φv3Ւ−1j−1+i=6i−1+j+4,i=1,2,…,Ւ−1,j=1,2,…n3,φvՒ−13j−2+i=6Ւ+n3+6i−1+j−8,i=1,2,…,Ւ−1,j=1,2,…,n3,φvՒ−13j−1+i=12Ւ+2n3+6i−1+j−20,i=1,2,…,Ւ−1,j=1,2,…,n3,φu3i=18Ւ+n+3i−28,i=0,1,…,n3−1,φu3i+1=18Ւ+2n+3i−28,i=0,1,…,n3−1,φu3i+2=18Ւ+3n+i−28,i=0,1,…,n3−1,φw2i=18Ւ+4n+5i−25,i=0,1,…,n2−1,φw2i+1=18Ւ+4n+5n2+5i−25,i=0,1,…,n2−1,φwn=1.
See Figure3(a).
Then, to showφ is a valid radio labelling, we must verify the inequality(5)da,b+φa−φb≥7∀a,b∈VCSn,Ւ.
Leta,b∈VCSn,Ւ.Case 1: suppose thata=vα and b=vβ,1≤α≠β≤nՒ−1.Case 1.1: ifα=3Ւ−1s−1+pandβ=3Ւ−1t−1+q or α=Ւ−13s−2+pandβ=Ւ−13s−2+p or α=Ւ−13s−1+pandβ=Ւ−13s−1+p where p≠q and s=t, then dvα,vβ=1 and φvα−φvβ≥6p−q≥6. In the same subcase, if s≠t, then dvα,vβ=6 and φvα−φvβ≥1. So, da,b+φa−φb≥7.Case 1.2: ifα=3Ւ−1s−1+pandβ=3Ւ−1t−1+q,where1≤p,q≤Ւ−1, 1≤s,t≤n/3, then φvα=6p−1+s+4andφvβ=6Ւ+n/3+6q−1+t−8. In addition, dvα,vβ≥4. Therefore, da,b+φa−φb≥4+6p−1+s+4−6Ւ+n/3+6q−1+t−8≥4+3=7.Case 1.3: ifα=3Ւ−1s−1+pandβ=Ւ−13t−1+q,where1≤p,q≤Ւ−1, 1≤s,t≤n/3, then φa−φb=6p−1+s+4−12Ւ+2n/3+6q−1+t−20>6. Consequently, the condition is true.Case 1.4: ifα=Ւ−13s−2+pandβ=Ւ−13t−1+q,where1≤p,q≤Ւ−1, 1≤s,t≤, then dvα,vβ≥4 and φvα−φvβ=6Ւ+n/3+6p−1+s−8−12Ւ+2n/3+6q−1+t−20≥3. It follows that da,b+φa−φb≥7.Case 2: takea=vα and b=vβ,0≤α,β≤n−1. If α=3s+pandβ=3t+q,0≤p=q≤2, 0≤s,t≤n/3−1, then dv3s+p,v3t+p=4 and φv3s+p−φv3t+p≥3. Otherwise, that is, if p≠q, thus, dv3s+p,v3t+q≥2 and φv3s+p−φv3t+q≥n>6sincen>6. Therefore, in both chances, we get da,b+φa−φb≥7.Case 3: assume thata=u2s+p and b=u2t+q,0≤s≤⌈n/2⌉−1,0≤t≤⌊n/2⌋−1,0≤p,q≤1.Ifp=q, then φu2s+p−φu2t+q≥4 and dv3s+p,v3t+p=3; else, dv3s+p,v3t+q≥2 and φv3s+p−φv3t+q≥⌈n/2⌉+⌊n/2⌋. Therefore, in both of them, the inequality is satisfied.Case 4: suppose thata=vα and b=uβ,0≤α≤nՒ−1, 0≤β≤n−1; then, either da,b=3 and φa−φb≥4 or da,b=1 and φa−φb>6. Therefore, da,b+φa−φb≥7.Case 5: if we takea=vα, 1≤α≤nՒ−1, and b=wβ, 0≤β≤n−1, then from the mapping, φa−φb≥12Ւ+2n/3+6Ւ−2+n/3−20−18Ւ+4n−25φ>n>6, which verifies the condition trivially.Case 6: assume thata=uα and b=wβ, 0≤α,β≤n−1. If α=n−1 and β=0, then φuα−φwβ=18Ւ+3n+n/3−1−28−18Ւ+4n+5⌈n/2⌉−25=6 and duα,wβ=1. Otherwise, φuα−φwβ>6. So, duα,wβ+φuα−φwβ>6.Case 7: leta be the centre vertex of the wheel and b be any other vertex in the graph. If b=v1, then φwn−φv1=1−5=4andda,b=3. Otherwise, the condition is obviously true. Thus, φ is a valid radio labelling, and the vertex wn−1 is labelled with the maximum number 18Ւ+4n+5⌊n/2⌋+5⌈n/2⌉−1−25=18Ւ+9n−30. Hence, rnCSn,Ւ≤18Ւ+9n−30, n>4 and n≡0mod3.Theorem 12.
Forn>5 and n≡1mod3, the radio number of the star-sun graph satisfies rrSSn,Ւ≤20Ւ+12⌊n/3⌋+5n−4.Proof.
First, we define an injectionφ:VSSn,Ւ⟶N as follows:(6)φv3Ւj−1+i=5i−1+j+4,i=1,2,…,Ւ,j=1,2,…,n3,φvՒ3j−2+i=5Ւ+n3+5i−1+j−1,i=1,2,…Ւ,j=1,2,…,n3,φvՒ−13j−1+i=10Ւ+2n3+5i−1+j−4,i=1,2,…,Ւ,j=1,2,…,n3,φvՒn−1+i=15Ւ+3n3+5i−1−6,i=1,2,…,Ւ,φu3i=20Ւ+3n3+3i−7,i=0,1,…,n3,φu3i+1=20Ւ+6n3+3i−4,i=0,1,…,n3,φu3i+2=20Ւ+9n3+3i−1,i=0,1,…,n3−1,φw2i=20Ւ+12n3+5i+1,i=0,1,…,n2−1,φw2i+1=20Ւ+12n3+5n2+i+1,i=0,1,…,n2−1,φwn=1.
We omit the rest of the proof as it is similar to the one for Theorem11.
The other two cases in Theorems11 and 12 are also left to the reader.Theorem 13.
Forn>5, the radio number of the wheel-sun graph satisfies(7)rrWSn,Ւ≤15Ւ+n3+8n−4,n≡0mod319Ւ+n3+8n−4,n≡1mod3.19+n3+8n−5,n≡2mod3
Figure4 illustrates the case n≡2mod3. We omit the proof of this theorem.Figure 4
A radio labelling of complete-sun graphCS9,4 and wheel-sun graph WS8,5.


## 4. Conclusion

In this paper, we have presented the relation between the radio number and radial radio number. We have also defined and investigated the bounds for the same problems for the graphsCSn,Ւ, SSn,Ւ, andWSn,Ւ. For the graph fan-sun graph SSn,Ւ, the problem is still considered as an open research problem that needs further investigation. Since the method of finding the radial radio number and radio number of the fan-sun graph is similar to the previous theorem, it is still open to the interested researchers to do a further research work that can extend our results to identify more relations between the radio number and radial number by studying the same problem for interconnection networks.



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