#### 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.

Let be a connected graph with diameter and radius . For any integer , , radio coloring of G is an assignment of color (positive integer) to the vertices of such that , where is the distance between and in G. The biggest natural number in the range of is called the radio chromatic number of G, and it is symbolized by . The minimum number is taken over all such radio chromatic numbers of which is called the radio chromatic number, denoted by .

Čada et al. [4] proved that, for any distance graph we have

Recently, Bantva [5] improved this general lower bound. Based on different values, the radio chromatic number is classified into different problems.

This section deals with certain results which connect with for any connected graph .

Definition 1. The eccentricity of a vertex z, represented by in a connected graph , is the maximum distance from to any other vertex in G. That is, . The maximum eccentricity of the vertices of G is called the diameter of the graph, and it is symbolized by d or . In addition, the radius of graph G, symbolized by or , is the minimum eccentricity of the vertices of G.

Definition 2. A connected graph is called a self-centred graph if In other words, .
The following is a straight result from the definitions of the radio number and radial radio number.

Theorem 1. For any connected graph ,.
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 is a connected graph of order and diameter , then .

Theorem 3. For a complete partite graph of order , .

Theorem 4. Every connected graph of order with is self-centred.

Using Theorem 5 and Definition 2, we have attained the equality of Theorem 1 as follows.

Theorem 5. A connected graph of order is self-centred if and only if .

Theorem 6. Let be a complete partite graph of order ; then, .

Proof. Let the vertex set of be partitioned into disjoint sets such that , and . The radius of the complete partite graph is 1, and all the vertices in the sets , are at distance two. Hence, we can label the vertices in each set as . Clearly, the radial radio labelling condition is satisfied for any pair of vertices in . Hence, .

Theorem 7. If is a connected graph of order and radius , then .

Proof. Given 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 by in Theorem 2. Consequently,

#### 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 graph complete-sun graph wheel-sun graph , and fan-sun graph .

Definition 3. A sunflower graph consists of a wheel with a centre vertex , -cycle , and additional vertices where is joined with edges to (), , and is taken as modulo . It is represented by The radius, diameter, and number of vertices of are 2, 4, and , respectively.

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

Definition 5. A star-sun graph, denoted by , is a graph obtained from the sunflower graph and copies of star graph by merging the internal vertex of the star graph and vertex of , , as shown in Figure 1(a).

Remark 1. The cardinality of and in is and , respectively. Also, the diameter and radius of the graph are 6 and 3, respectively.

Definition 6. A complete-sun graph, denoted by , is a graph obtained from the sunflower graph and copies of complete graph by merging a vertex of the complete graph and the vertex of , , as shown in Figure 1(b). Here, we have and .

Remark 2. The diameter and radius of are 6 and 3, respectively.

Definition 7. A wheel-sun graph, denoted by , is a graph obtained from the sunflower graph and copies of wheel graph by merging the vertex of and the centre vertex of the wheel, where as shown in Figure 1(c).

Remark 3. The number of vertices in is , while its number of edges is . Also, its diameter and radius are 6 and 3, respectively.

Definition 8. A fan-sun graph is a graph obtained from the sunflower graph and copies of fan graph by merging of the fan and the vertex of , . It is denoted by as shown in Figure 1(d).

Remark 4. For the graph the number of edges is , while the number of vertices is . Moreover, the diameter and radius are 6 and 3, respectively.
In this work, we name the newly included vertices of , , and as in the clockwise sense.

The following theorems provide the upper bound for the radial radio number of , .

Theorem 8. Let G be the sun-star graph Then, .

Proof. First, we define a mapping as follows: , , , , , and as shown in Figure 2.
Since the radius of the graph is 3, we must verify satisfies the radial radio labelling condition for every pair of vertices .

Theorem 9. Let G be the complete-sun graph . If then the radial radio number of satisfies

Proof. Let us name the newly included vertices of as in the clockwise sense. Now, we define a one-one mapping as follows:This mapping is visible in Figure 3(a).
In the following, we claim that .
Let .Case 1: suppose and , , then and Case 1.1: if , then and Therefore, since .Case 1.2: if , then , which is enough for verifying the condition.Case 2: assume that , , , and ; then, and . Also, Therefore, .Case 3: if we take , , , and , then since , which verifies the condition trivially.Case 4: assume that and Case 4.1: if and , then and However, . Therefore, . Otherwise, is greater than 3.Case 5: let and , . In this case, if and or and or and , 0 then and Otherwise, and Hence, in both possibilities, the condition for radial radio labelling is satisfied.Case 6: assume that and , . If and or and , , , then . Thus, and Otherwise, and Therefore, the condition holds in both of the possibilities.Case 7: finally, let us assume that , and is any vertex in . If then and Otherwise, the radial radio labelling condition is obviously true. Thus, satisfies the condition of radial radio labelling and attains the maximum value for the vertex . Therefore, we get The proof for the other two cases, namely, and , is left to the reader.

Theorem 10. For , the radial radio number of the wheel-star graph satisfiesWe omit the proof, but Figure 3(b) illustrates the case .

##### 3.2. Radio Number of Sunflower Extended Graphs

This section provides the upper bound for the radio number of , .

Theorem 11. For and the radio number of the complete-sun graph satisfies

Proof. We define a 1-1 mapping as follows:See Figure 3(a).
Then, to show is a valid radio labelling, we must verify the inequalityLet .Case 1: suppose that and Case 1.1: if or or where and , then and . In the same subcase, if then and So, Case 1.2: if , , then . In addition, . Therefore, Case 1.3: if , , then Consequently, the condition is true.Case 1.4: if , , then and . It follows that .Case 2: take