Radial Radio Number of Hexagonal and Its Derived Networks

A mapping ℸ : V ( G ) ⟶ N ⋃ 0 { } for a connected graph G � ( V, E ) is called a radial radio labelling if it satisﬁes the inequality | ℸ ( x ) − ℸ ( y )| + d ( x, y ) ≥ rad ( G ) + 1 ∀ x, y ∈ V ( G ) , where rad ( G ) is the radius of the graph G . The radial radio number of ℸ denoted by rr ( ℸ ) is the maximum number mapped under ℸ . The radial radio number of G denoted by rr ( G ) is equal to min { rr ( ℸ ) / ℸ is a radial radio labelling of G


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. is 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: V(G) ⟶ N such that d(x, y) + |ℸ(x) − ℸ(y)| ≥ 1 + diam(G)∀x, y ∈ V(G). e radio number of ℸ, denoted by rn(ℸ), is the maximum number assigned to any vertex of G. e radio number of G, denoted by rn(G), is the minimum value of rn(ℸ) 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 |ℸ(x) -ℸ(y)| as (ℸ(x) + ℸ(y))/2, ��������� ℸ(x)ℸ(y), |ℸ(x)ℸ(y)|/2, and (2ℸ(x)ℸ(y))/(ℸ(x) + ℸ(y)) or diam(G) as diam(G) − 1, rad(G), 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: is called a radial radio labelling if this satisfies the inequality |ℸ(x) − ℸ(y)| + d(x, y) ≥ rad(G) + 1 ∀x, y ∈ V (G), where rad(G) is the radius of the graph G. e radial radio number of ℸ denoted by rr(ℸ) is the maximum number mapped under ℸ. e radial radio number of G denoted by rr(G) is equal to min {rr(ℸ)/ℸ is a radial radio labelling of G}. It is obvious from the definitions that, for any connected graph G, rn(G) ≥ rr(G). However, the radial radio number is reduced to the radio number for any selfcentered graphs, which is for the graphs that satisfy rad(G) � diam(G). For example, the radio number and radial radio number for the complete graphs K n and complete bipartite graphs K m,n are n and m + n + 1, respectively. Hence, rn(G) > rr(G) for any graph G which is not self-centered. Selvam et al. [4] proved that rr(G) ≥ n for any self-centered graph G. In addition, Selvam et al. [5] proved few results connecting the clique number ω and rr(G) as follows: (i) ω(G) ≤ rr(G), (ii) for m ≥ 1, ∃ a graph G which satisfies rr(G) � m + ω and ω � 3, and (iii) ∃ a graph G with rr(G) � ω + 1, whenever ω ≥ 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 rr(MT(n)) ≤ 2n + 1 and rr(D(n)) ≤ 2n + 2, where MT(n) is the Mongolian tent and D(n) is the diamond graph. is research article highlights the newly defined enhanced hexagonal difference hexagonal network. Also, the radial radio number for HX η , HDN(η), and EDH(η) was resolved.

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][9][10]. In the graph theoretical approach, a hexagonal network of dimension η is denoted by HX η which contains 6(η − 2) vertices of degree 4, 6 corner vertices of degree 3, and 3η 2 − 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. erefore, diam(HX η ) and rad(HX η ) are 2η − 2 and η − 1, respectively. In addition, it has 9η 2 − 15η + 6 edges and 3η 2 − 3η + 1 vertices. See Figure 1(a).
Even though the vertices of the three axes α, β, and c of HX η 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 y 1 1 , y 1 2 , . . . , y 1 κ , y 2 1 , y 2 2 , . . . , y 2 η+1 , . . . , y . Furthermore, the face vertices of HDN(η) are named in the same manner from the left most top to the right most bottom as

Radial Radio Number of HX η , HDN(η), and EDH(η)
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 network EDH(η).

Theorem 2. Let HDN(η) be an enhanced hexagonal network of dimension κ; then, the radial radio number of HDN(η) satisfies rr(HDN(η))
Proof. First, let us partition the face vertices of HDN(η) into four disjoint sets e remaining vertices in HDN(η) are partitioned into 5 disjoint sets as in eorem 1. Now, we define a mapping ℸ: V (HDN(η)) ⟶ N ∪ 0 { } as follows: 4 International Journal of Mathematics and Mathematical Sciences (3) e remaining vertices are labelled as in eorem 1. See Figure 3(b).

Conclusion
In this research study, we have introduced the enhanced hexagonal difference hexagonal network from the existing enhanced hexagonal network. Furthermore, we have investigated rr(EDH(η) ) ≤ (2η − 3) η(η − 2) + 2(η − 1) 2 + 1, η > 1. Also, the radial radio number of HDN(η) and EDH(η) was presented. ese 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.

Data Availability
No data were used to support this study.

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