Radio and Radial Radio Numbers of Certain Sunflower Extended Graphs

of , assignment color to vertices that d a, b ) + φ a φ ( b 1 + k , ∀ a, b ∈ V ( G ) where d ( a, b ) is between a and b in number in range called the radio − chromatic number of it symbolized by r ck ( φ . such radio − chromatic numbers of which called the radio − chromatic denoted by r ck ( G ) . For k � d and k � ρ , radio k − numbers are termed as the radio number ( rn ( G ) ) and radial radio number ( rr ( G ) ) of , respectively. In this research work, the relationship between the radio number and radial number is studied for connected graph. Then, several sunﬂower extended graphs are deﬁned, and the upper bounds of the radio number and radial radio number are investigated


Introduction
e 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. erefore, 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-kchromatic number as follows.
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 d(a, b) is the distance between a and b in G. e biggest natural number in the range of φ is called the radio k− chromatic number of G, and it is symbolized by r ck (φ). e minimum number is taken over all such radio k− chromatic numbers of φ which is called the radio k− chromatic number, denoted by r ck (G).
Cada et al. [4] proved that, for any distance graph D(t − 1, t), we have Recently, Bantva [5] improved this general lower bound. Based on different k values, the radio k− chromatic number is classified into different problems.
For k � d, the radio k− chromatic number is termed as the radio number problem, and it is symbolized by rn(G). 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. e 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: V(G) ⟶ N ∪ 0 { } for a connected graph G � (V, E) is called a radial radio labelling if this satisfies the inequality |φ(a) − φ(b)| +d(a, b) ≥ ρ +1∀a, b ∈ V(G), where ρ is the radius of the graph G. Radial radio number of φ symbolized by rr(φ ) is the maximum number mapped under φ. e radial radio number of G, denoted by rr(G), is equal to min rr(φ) /φ is a radial radio labelling of G}. 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 rn(G) and rr(G). Furthermore, we have defined and determined the radio and radial radio numbers of certain sunflower extended graphs such as SS(n, Ւ), CS(n, Ւ), and WS(n, Ւ).

Relation between the Radio Number and Radial Radio Number
is section deals with certain results which connect rn(G) with rr(G) for any connected graph G.

Definition 1.
e eccentricity of a vertex z, represented by e(z) in a connected graph G, is the maximum distance from z to any other vertex in G.
}. e maximum eccentricity of the vertices of G is called the diameter of the graph, and it is symbolized by d or di am(G). In addition, the radius of graph G, symbolized by ρ or ra d(G), is the minimum eccentricity of the vertices of G.
e following is a straight result from the definitions of the radio number and radial radio number.

Theorem 1. For any connected graph G, rn(G) ≥ rr(G).
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.
If G is a connected graph of order n and diameter d, then n ≤ rn(G) ≤ (n − 1)d.

Theorem 4.
Every connected graph G of order n with rn(G) � n is self-centred.
Using eorem 5 and Definition 2, we have attained the equality of eorem 1 as follows.

Theorem 5.
A connected graph G of order n is self-centred if and only if rn(G) � rr(G) � n. Theorem 6. Let G � (V, E) be a complete k− partite graph of order n; then, rr(G) � k.
e radius of the complete k− partite graph is 1, and all the vertices in the sets U i , 1 ≤ i ≤ k, are at distance two. Hence, we can label the vertices in each set U i as i(i � 1, 2, . . . , k). Clearly, the radial radio labelling condition d(a, b) + |φ(a) − φ(b)| ≥ 2 is satisfied for any pair of vertices in G. Hence, rr(G) � k. □ Theorem 7. If G is a connected graph of order n > 1 and radius ρ, then 2 ≤ rr(G) ≤ (n − 1)ρ.
Proof. Given G is a connected graph that contains at least two vertices. erefore, the lower bound of the theorem attains in the particular case of eorem 6 which is for the complete bipartite graphs. Furthermore, the upper bound is obtained by replacing d by ρ in eorem 2. Consequently, 2 ≤ rr(G) ≤ (n − 1)ρ, n > 1.

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 graph SS(n, Ւ), complete-sun graph CS(n, Ւ), wheelsun graph WS(n, Ւ), and fan-sun graph FS(n, Ւ).

Definition 3.
A sunflower graph consists of a wheel with a centre vertex w n , n-cycle w 0 , w 1 , . . . , w n− 1 , and additional n vertices u 0 , u 1 , . . . , u n− 1 where u i is joined with edges to (w i , w i+1 ), i � 0, 1, 2, . . . n-1, and i + 1 is taken as modulo n. It is represented by SF n . e radius, diameter, and number of vertices of SF n are 2, 4, and 2n + 1, respectively.

Definition 4.
A star graph, denoted by S Ւ+1 , is defined as a complete bipartite graph of the form K 1,Ւ , Ւ > 1. In other words, S Ւ+1 is a tree having Ւ leaves and one internal vertex.

Definition 5.
A star-sun graph, denoted by SS(n, Ւ), is a graph obtained from the sunflower graph SF n and n copies of star graph S Ւ+1 by merging the internal vertex of the k th star graph S Ւ+1 and vertex u k− 1 of SF n , 1 ≤ k ≤ n, as shown in Figure 1(a).

Definition 7.
A wheel-sun graph, denoted by WS(n, Ւ), is a graph obtained from the sunflower graph SF n and n copies of wheel graph W Ւ+1 by merging the vertex u k− 1 of SF n and the centre vertex of the k th wheel, where 1 ≤ k ≤ n as shown in Figure 1(c).

Definition 8.
A fan-sun graph is a graph obtained from the sunflower graph SF n and n copies of fan graph F Ւ+1 � P Ւ + K 1 by merging K 1 of the k th fan and the vertex u k− 1 of SF n , 1 ≤ k ≤ n. It is denoted by FS(n, Ւ) as shown in Figure 1(d).

Radial Radio Number of Sunflower Extended Graphs.
e following theorems provide the upper bound for the radial radio number of S(n, Ւ), CS(n, Ւ), and WS(n, Ւ).
Since the radius of the graph is 3, we must verify φ satisfies the radial radio labelling condition d( Let us choose any two arbitrary vertices a and b in the sun-star graph. respectively. Also, a and b are at a distance two. Hence, the radial radio labelling condition becomes d( then a and b are at a distance at least 4. Hence, the radial radio labelling condition is trivially satisfied.
which trivially verifies the radial radio labelling condition.
Case 5: if a is the centre vertex of the wheel and b is any other star vertex, then the distance between them is exactly 3. Also, φ(a) � 0 and φ Case 6: let a and b be the vertices in the n-cycle of the sunflower graph.
, which is enough for verifying the condition.

Radio Number of Sunflower Extended Graphs.
is section provides the upper bound for the radio number of S(n, Ւ), SS(n, Ւ), and WS(n, Ւ).
Case 3: assume that a � u 2s+p and b � u 2t+q , . erefore, in both of them, the inequality is satisfied.
Case 7: let a be the centre vertex of the wheel and b be any other vertex in the graph.

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 graphs CS(n, Ւ), SS(n, Ւ), and WS(n, Ւ). For the graph fan-sun graph SS(n, 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.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Authors' Contributions
Mohammed K. A. Kaabar contributed to actualization and initial draft, provided the methodology, validated and investigated the study, supervised the original draft, and edited the article. Kins Yenoke validated and investigated the study, provided the methodology, performed formal analysis, and contributed to the initial draft. Both authors read and approved the final version.