Graph theory is related with numerous divisions of mathematics like group theory, matrix theory, numerical analysis, probability and topology. It has also applications in other scientific subjects such as physics, chemistry, communication science, computer technology, electrical engineering, and so on. Graph theory is also associated with the field of chemical graph theory.

There are many reasons for the increasing popularity of graph theory in chemistry. Chemical graph theory is a subdivision of mathematical chemistry in which points denote atoms and lines denote chemical bonds in the underlying chemical structure. Graph theory offers simple rules by which chemists obtain qualitative predictions about the structure and reactivity of various compounds. Graph theory is used as a basis for the representation, classification, and categorization of a very huge chemical system and carbon Nano structures. In the field of chemistry, graph theory is more useful theoretical tool. The study of fuzzy radio reciprocal L- labeling is applicable to chemical graphs like V-Phenylenic Nanosheets, Sirpinski graphs. In fuzzy radio reciprocal L- labeling of a chemical graph, weight of points and lines are assigned from the interval (0,1). L is the length of any two pair of points and it has the main role to assign weights to every other of the points to reach the goal.

Hale (1980) formulated graph vertex coloring problem based on the frequency assignment problem where a frequency is assigned to each radio transmitter

Gani & Radha (2009) found the degree of vertices in union of two fuzzy graphs, sum of two fuzzy graphs and Cartesian product of two fuzzy graphs

In general, the focus on radio labeling is to avoid interference between any two radio stations. If the distance between any two radio stations is close, then the interference is unavoidable. In 2005, Chartrand et al. introduced the concept of radio labeling and put the level of interference at largest possible-the diameter of graph. For any graph, points are labeled by the positive integers and then the sum of the difference between of any two labeled points a and b and distance between a and b is greater than or equal to diameter of graph. Even if distance is close, then difference between the values assigned to the concerned radio channels is high. In radio labeling distance depends on the number of minimum lines between any two radio stations

In radio channel assignment problems, the channel frequency difference between two channels is 0.2MHz. For broadcasts in air, the available bandwidth is determined by the allocated frequency band. For example, AM radio signals must be in the 535-1605 kHz band. Therefore, there are 107 possible carrier frequencies that may be assigned, at 10 kHz intervals. Likewise FM radio stations are at 0.2 MHz intervals in the range of 88-108 MHz allowing 100 stations in a local area

A fuzzy graph

where

V-Phenylenic nanotubes VPH(m, n) is called fuzzy V-Phenylenic N-tubes FVPH(m, n),

Proof: Let FCHG be any n- dimensional chemical graph with points

Therefore the weight of points and lines lie between

Proof: Let us take any n- dimensional fuzzy V-Phenylenic Nano sheet, assign the weight to the lines which are always less than the corresponding weights of the points. When weights are allotted to the lines in FVPH(m ,n) then the reciprocal the weight of any weight of the lines must be 1 or 1.xx…x where x varies from 1 to 9.

When m

The sum of the number of lines on the boundary of

When m < n

The sum of the number of lines on the boundary of

Proof: In an n-dimensional Sirpinski graph (n , k),

Proof: Let VPH(m, n) be the n-dimensional Nano sheet with points

Proceeding similar way assign weights to points by

From

To prove: FVPH(m, n) satisfies

Let

When m

Case (i): When

Case (ii): Assign weights to points row-wise

Case (iii): when

Case (iv) when

When m < n the following cases hold true, Assign weight sof points column-wise

Case (i): When

Case (ii): Assign weights of points row wise

When

Case (iii): when

Case (iv): when

Step1: Assign weights to lines by

If

Step 2: Assign weights to points by

If m

If m < n, assign weights to points column-wise.

Step 3: Find

(

Step 4: Find Diam(FVPH(m, n)) using Step 3. If Diam(FVPH(m, n)) then go to step 5.

If Diam(FVPH(m, n)) then go to step 1 and rearrange weights of

Step 5: The highest of weight of the points is

Sirpinski gasket graph (S_{n}) is called Fuzzy Sirpinski gasket graph

Proof: Let

When l = 1, the _{n-}_{1}, T, (S_{n-}_{1}, L), (S_{n-}_{1}, R)) which gives the minimum possible difference of _{n-}_{1}, T, (S_{n-}_{1}, L), (S_{n-}_{1}, R))

_{n}

Step 1: Assign weights to points from (S_{n-}_{1}, T) to (S_{n-}_{1}, ) (S_{n-}_{1}, R) by

Step 2: Assign weights to lines from (S_{n-}_{1}, T) to (S_{n-}_{1}, L) (S_{n-}_{1}, R) by

If

Step 3: Find

by step 2

Step 4: Find diam(S_{n}) using Step 3. If

If diam(S_{n}) >

Step 5: The highest of weights of the points is

Sirpinski Like graph (S(n,4)) is called Fuzzy Sirpinski graph

Proof: Let S(n,4) be the n-dimensional Nano sheet with points

Proceeding in a similar way assign weights to lines by

From above equations,

Case(i): If L=1 and

Case(ii): If L=1 then

For

Proceeding in this way

Let (

In this case

When

Similarly for n – dimensional S(n,4), n times of (

By lemma 5.1.2 diam(S(n,4)) <

Case(iii) If

By (case i) and (case ii) it satisfies

Step 1: Assign weights to points clockwise in every copy of

Step 2: Assign weights to lines clockwise in every copy of

If

Step 3: Find

Step 4: Find diam(S(n ,4)) by using Step 3. If diam(S(n ,4)) <

If diam(S(n ,4)), then go to step 1 and rearrange weights of

Step 5: The highest of weight of the points is

There are many routers with minimum dimensions and minimum distance which make more interference between any Wi-Fi routers. In this model of chemical graph, Wi-Fi routers are very close with minimum circumference area. Some of the routers have very minimum distance and maximum difference connected by one line. Assign frequencies to the routers by fuzzy weights from the interval 0 and 1. If the weight of every router lies between (0.5xxx...x GHz, 0.9xxx...x GHz) then the signal is very strong. WD- Wi-Fi router capacity difference, DS- Distance between Routers, L- Number of lines in between routers, I- Interference, SS- Signal Strength. Here the of WD and DS are given by assigning two decimals of fuzzy weights. Tables values of WD and DS varies depending on their decimals weights.

Let A, B, C, D, E, F are six Wi-Fi routers connected by line. Assign weights to lines and points from (0.5xxx, 0.9xxx). Here consider weights of points as Wi-Fi router frequencies and the minimum

From

x =WD(GHz) |
d= DS(KM) |
L |
D=3.96 |

0.01GHz |
2.00km, 1.96km, 1.92km, 1.88km, 1.85km, 1.81km, 1.78km, 1.75km, 1.72km. |
1 |
d +1 |

0.01GHz |
3.96km, 3.80km, 3.79km, 3.68km, 3.53km, 3.47km |
2 |
d +2 |

If Wi-Fi routers are assigned based on fuzzy weights of figure 5.1, then there is no interference in between the routers and the strength of signals are good or excellent.

In this study, the methodology was introduced for fuzzy radio reciprocal L-labeling. Here the general formula of fuzzy radio reciprocal L-labeling has been mathematically derived to apply this concept in chemical graphs with further result and discussions. In this connection the frequencies of Wi-Fi are assigned to the structure of chemical graphs. The generalized formula applied to certain chemicals graphs are N-tube, Sirpinski gasket graph and Sirpinski like graph satisfy fuzzy radio reciprocal L-labeling where every frequency of channels is from the closed interval 0 to 1.