Edge Weight Based Entropy Measure of Different Shapes of Carbon Nanotubes

Entropy in general, is termed as the measure or determination of the improbability of a system. This opulent idea has invited wide interest while discussing and dealing with the multidimensional aspects of applied mathematical chemistry. This idea, pertaining to graph theory, was first applied in 1955. In the same way as nodes and edges create a graph of a network or system in mathematical chemistry, networks are built up in nodes and edges to form a graph of a network or system. Carbon nanotubes, on the other hand, are well-known for their application in memory devices as well as tissue engineering. This useful chemical structure, a carbon nanotube with one-end and both-end caps, is investigated. The entropy of these three carbon nanotube topologies is measured. Beside the analytical investigation of entropy measure of armchair carbon nanotubes, some comparison work is presented.


I. INTRODUCTION
"The entropy of a probability distribution known as a measure of the unpredictability of information content or a measure of the uncertainty of a system." This formulation was the foundation as described in [1] as a proposal for the idea of entropy. Following the statistical methodology of this concept, it became well-known for graphs and chemical structures. This parameter imparts a lot of information about structures, graphs, and chemical topologies. It was in 1955 that this term was used for graphs. Entropy, also known as graph-based entropy, has applications in chemistry, biology, ecology, sociology and a variety of other technical fields [2], [3]. Intrinsic and extrinsic measures are two types of entropy measurements based on graphs, that are computed by distinct elements (of graphs) and linked to probability distributions. From the network theory, there is a parameter known as degree-powers which is an application from graph theory mathematics to investigate networks as information functionals (reference [4], [5]). The authors in [6], put forward the idea of entropy for wide variety of networks. The entropy formulae put forward by [1], with the notion that knowing about the contents of a network and also for the structural information [7]. This notion was developed by using graphs, which greatly aided in learning more about and exploring biological systems. It has been used to investigate live organisms, for example, by creating a graph of chemical and biological systems. Given in [8], [9] for this biological and chemical application study. More information on entropy in computer sciences, structural chemistry, and in even biology can be found in [10] extensive study work. The entropy, which is explored in this study document, can be found in [7], [11] in network heterogeneity work. Nano-technology's electron transport capability, as well as its low-cost applications across a wide range of modern technology, has made nanostructure research a hot topic. Nanostructures were devised in 1990s, and throughout this decade, several other nano-variations were developed. After the discovery of nanotube in 1985 [12], carbon nanotubes were found in 1991 [13]. Because of the electron transport properties and material properties of carbon nanotubes, higherperformance and more developed electrochemical sensors are predicted in the foreseeable future, according to [14]. Carbon nanotubes are already used to make the most diverse electrochemical sensors, according to [15]. Because of the benefits of carbon nanotubes, besides electrochemical, sensors from biochemical fields are in the advanced stages of development [16]. Such sensors as are made up with biological techniques, are utilized to determine the proteins present in the human body and to check DNA, according to [17]. Such sensors can also detect cholesterol, glucose, lactate, pyrophosphate, alcohols, and a variety of other analytes, according to [16]. Biosensors can be developed and brought into function better than nanotubes. Nanotubes also help in biological identification in biosensor design. Carbon nanotubes are also used to immobilize the relevant chemical, according to [18]. There are three major forms of carbon nanotubes while looking at their structure from a topological perspective. Capped carbon nanotubes have both sides closed, while semi-capped carbon nanotubes have one side closed or open. See [19] for all these and many more topological characteristics of armchair carbon nanotubes. In literature, there are several studies and applications of carbon nanotubes. Following are some of the studies and applications of this structure. In [20], the quantum scale analysis of this new topology is described. Many technologies are used to identify nanotubes, but the detection of surface defects is investigated in [21]. It was utilized in photovoltaic [22] and saturable absorbers. In [23], the benefits of carbon nanotubes for this film are described, whereas in [24], the benefits of carbon nanotubes for fuel cells are highlighted. CVD development by palm oil pioneer is described in [25]. Nanotubes have also been used to enhance pharmaceutical applications, such as regenerative medications [26], [27]. It's also utilized to record dates and for the memory device [28] functionality. See [29], [30] for data on carbon nanotubes in biomedical engineering, particularly in tissue function. Solar cell coating nanotubes are also employed as a key element in anti-reflection, according to [31]. The definitions of topological indices that we used to discover our major conclusions and create formulae for edge weight based entropy are listed below. Definition 1: The first and second Zagreb index is introduced in 1972 by Gutman [32], [33] as: Definition 2: Shirdel et.al [34] introduced Hyper-Zagreb index as: Definition 3: Furtula et al. [35] defined Augmented Zagreb index as: Definition 4: Ranjini et al. [36], defined, the redefined first, second and third Zagreb indices for a graph G as; Definition 5: In 2014, the entropy for an edgeweighted graph was introduced [37]. Let a graph G = (V (G) , E (G) , ψ (ab)) , with V (G) denotes the vertex set, E (G) stand for edge set and the important or new factor is ψ (ab) is a weight for an edge ab. Now following is the main formula of this research work is the entropy for an edge weighted graph; The researchers in [38], [39] established the following entropies for an edge weighted based graph by making the edge of a weight equal to the major portion of the topological index. The following are some key formulas for this study, all of which are based on the Equation (8). Definition 6: The first and second Zagreb entropies are defined as following [38], [40] ; Definition 7: The hyper and augmented Zagreb entropies are defined as following [38] ; Ω AZI (G) Definition 8: The first, second and third redefined Zagreb entropies are defined as following [39] ; Ω ReZG3 (G) The major contribution of this paper is to broaden the range of uses for armchair carbon nanotubes by investigating various edge weight-based entropies. The notions ACN T (β, γ) , ACSCN T (β, γ) and ACCN T (β, γ) , are armchair carbon semi-capped nanotubes and armchair carbon capped nanotubes, respectively. Furthermore, we analyzed these entropies and compared the three stated structures using instances from Table 9 to Table 15 in the next section, and presented the findings in Figure 5 to Figure 11. The article [41] has additional information on this structure of armchair carbon nanotube and its variants.

II. RESULTS FOR THE UNCAPPED CARBON NANOTUBE ACN T (β, γ)
The notion ACN T (β, γ) , is for an armchair carbon nanotube, it has two types of vertices according to degree defined in Table 1 and edge types defined in Table 2, and p 1 are the order and q 1 , size of ACN T (β, γ) .
Authors of [42], determined the topological indices based on the degree of armchair carbon nanotube ACN T (β, γ) . We defined the main results in Table 3, we will used these results in our main prove of theorems. Furthermore, a three-dimensional image of an armchair carbon nanotube ACN T (β, γ) is shown in Figure 1.
Theorem 9: Let Ω M1 is the edge weight based first Zagreb entropy for the graph G ∼ = ACN T (β, γ) , then Ω M1 (G) is Proof. By using the Table 2 given for the edge varieties of ACN T (β, γ) graph, first Zagreb index is determined in the Table 3. Now for the entropy of first Zagreb, we will use the the value of first Zagreb index in the Equation 9, and resulted FIGURE 1: Three-dimensional sight of armchair carbon nanotube ACN T (β, γ) VOLUME 4, 2020 in 2 Theorem 10: Let Ω M2 is the edge weight based second Zagreb entropy for the graph is Proof. By using the Table 2 given for the edge varieties of ACN T (β, γ) graph, second Zagreb index is determined in the Table 3. Now for the entropy of second Zagreb, we will use the the value of second Zagreb index in the Equation 10, and resulted in 2 Theorem 11: Let Ω HM is the edge weight based hyper Zagreb entropy for the graph G ∼ = ACN T (β, γ) , then Proof. By using the Table 2 given for the edge varieties of ACN T (β, γ) graph, hyper Zagreb index is determined in the Table 3. Now for the entropy of hyper Zagreb, we will use the the value of hyper Zagreb index in the Equation 11, and resulted in 2 Theorem 12: Let Ω AZI is the edge weight based augmented Zagreb entropy for the graph G ∼ = ACN T (β, γ) , then is Proof. By using the Table 2 given for the edge varieties of ACN T (β, γ) graph, augmented Zagreb index is determined in the Table 3. Now for the entropy of augmented Zagreb, we will use the the value of augmented Zagreb index in the Equation 12, and resulted in 2 Theorem 13: Let Ω ReZG1 is the edge weight based redefined first Zagreb entropy for the graph G ∼ = ACN T (β, γ) , then Proof. By using the Table 2 given for the edge varieties of ACN T (β, γ) graph, redefined first Zagreb index is determined in the Table 3. Now for the entropy of redefined first Zagreb, we will use the the value of redefined first Zagreb index in the Equation 13, and resulted in Proof. By using the Table 2 given for the edge varieties of ACN T (β, γ) graph, redefined second Zagreb index is determined in the Table 3. Now for the entropy of redefined second Zagreb, we will use the the value of redefined second Zagreb index in the Equation 14, and resulted in 2

Theorem 15:
Let Ω ReZG3 is the edge weight based redefined third Zagreb entropy for the graph G ∼ = ACN T (β, γ) , then Proof. By using the Table 2 given for the edge varieties of ACN T (β, γ) graph, third redefined Zagreb index is determined in the Table 3. Now for the entropy of redefined third Zagreb, we will use the the value of redefined third Zagreb index in the Equation 15, and resulted in 2

III. RESULTS FOR THE UNCAPPED CARBON NANOTUBE ACSCN T (β, γ)
The notion ACSCN T (β, γ) , is for the armchair carbon semi-capped nanotub, which has two different types of vertices, according to the fact established in Table 4, while edge distribution is defined in Table 5, given p 2 , q 2 , are the order and size of ACSCN T (β, γ) , respectively. The data of the topological indices for the structure of ACSCN T (β, γ) , namely armchair semi-capped nanotube, given in [43]. The main results of this research work is useful here and we summarized in the Table 6, and we will use these results in our main prove of theorems. Moreover, the 3D view of armchair semi-capped nanotube ACSCN T (β, γ) is shown in Figure 2. Theorem 16: Let Ω M1 is the edge weight based first Zagreb entropy for the graph G = ∼ = ACSCN T (β, γ) , then Ω M1 (G) is   ACSCN T (β, γ) .
Proof. By using the Table 5 given for the edge varieties of ACSCN T (β, γ) graph, first Zagreb index is determined in the Table 6. Now for the entropy of first Zagreb, we will use the the value of first Zagreb index in the Equation 9, and resulted in Theorem 17: Let Ω M2 is the edge weight based second Zagreb entropy for the graph G = ∼ = ACSCN T (β, γ) , then Proof. By using the Table 5 given for the edge varieties of ACSCN T (β, γ) graph, second Zagreb index is determined in the Table 6. Now for the entropy of second Zagreb, we will use the the value of second Zagreb index in the Equation 10, and resulted in 2 Theorem 18: Let Ω HM is the edge weight based hyper Zagreb entropy for the graph G = ∼ = ACSCN T (β, γ) , then Ω HM (G) is Proof. By using the Table 5 given for the edge varieties of ACSCN T (β, γ) graph, hyper Zagreb index is determined in the Table 6. Now for the entropy of hyper Zagreb, we will use the the value of hyper Zagreb index in the Equation 11, and resulted in Proof. By using the Table 5 given for the edge varieties of ACSCN T (β, γ) graph, augmented Zagreb index is determined in the Table 6. Now for the entropy of augmented Zagreb, we will use the the value of augmented Zagreb index in the Equation 12, and resulted in Proof. By using the Table 5 given for the edge varieties of ACSCN T (β, γ) graph, redefined first Zagreb index is determined in the Table 6. Now for the entropy of redefined first Zagreb, we will use the the value of redefined first Zagreb index in the Equation 13, and resulted in Proof. By using the Table 5 given for the edge varieties of ACSCN T (β, γ) graph, redefined second Zagreb index is determined in the Table 6. Now for the entropy of redefined second Zagreb, we will use the the value of redefined second Zagreb index in the Equation 14, and resulted in

Theorem 22:
Let Ω ReZG3 is the edge weight based third redefined Zagreb entropy for the graph G = ∼ = ACSCN T (β, γ) , Proof. By using the Table 5 given for the edge varieties of ACSCN T (β, γ) graph, redefined third Zagreb index is determined in the Table 6. Now for the entropy of redefined third Zagreb, we will use the the value of redefined third Zagreb index in the Equation 15, and resulted in 2

IV. RESULTS FOR THE UNCAPPED CARBON NANOTUBE ACCN T (β, γ)
The notion ACCN T (β, γ) , is for the armchair carbon capped nanotube, it has a solemn type of vertices and edge types defined in Table 7, along the fact p 3 , q 3 , are the order and size of ACCN T (β, γ) , respectively. The data on the topological indices based on the degree of graph ACCN T (β, γ) or armchair capped nanotube given in [43]. For the usage of that data we summarized in the Table  8, and we will take into account for the results of our main prove of theorems. Furthermore, a three-dimensional image of an armchair carbon capped nanotube ACCN T (β, γ) is shown in Figure 3. While two-dimensional symmetry of this structure is shown in Figure 4 Theorem 23: Let Ω M1 is the edge weight based first Zagreb entropy for the graph G = ∼ = ACCN T (β, γ) , then Ω M1 (G) is    Proof. By using the Table 7 given for the edge varieties of ACCN T (β, γ) graph, first Zagreb index is determined in the Table 8. Now for the entropy of first Zagreb, we will use the the value of first Zagreb index in the Equation 9, and resulted in 2

Theorem 24:
Let Ω M2 is the edge weight based second Zagreb entropy for the graph G = ∼ = ACCN T (β, γ) , then Ω M2 (G) is Proof. By using the Table 7 given for the edge varieties of ACCN T (β, γ) graph, second Zagreb index is determined in the Table 8. Now for the entropy of second Zagreb, we will use the the value of second Zagreb index in the Equation 10, and resulted in is Proof. By using the Table 7 given for the edge varieties of ACCN T (β, γ) graph, hyper Zagreb index is determined in the Table 8. Now for the entropy of hyper Zagreb, we will use the the value of hyper Zagreb index in the Equation 11, and resulted in 2

Theorem 26:
Let Ω AZI is the edge weight based augmented Zagreb entropy for the graph G = ∼ = ACCN T (β, γ) , then Ω AZI (G) is Proof. By using the Table 7 given for the edge varieties of ACCN T (β, γ) graph, augmented Zagreb index is determined in the Table 8. Now for the entropy of augmented Zagreb, we will use the the value of augmented Zagreb index in the Equation 12, and resulted in Proof. By using the Table 7 given for the edge varieties of ACCN T (β, γ) graph, first redefined Zagreb index is determined in the Table 8. Now for the entropy of first redefined Zagreb, we will use the the value of first redefined Zagreb index in the Equation 13, and resulted in 2

Theorem 28:
Let Ω ReZG2 is the edge weight based second redefined Zagreb entropy for the graph G = ∼ = ACCN T (β, γ) , then Ω ReZG2 (G) is Proof. By using the Table 7 given for the edge varieties of ACCN T (β, γ) graph, second redefined Zagreb index is determined in the Table 8. Now for the entropy of second re-defined Zagreb, we will use the the value of second redefined Zagreb index in the Equation 14, and resulted in (55) 2

Theorem 29:
Let Ω ReZG3 is the edge weight based third redefined Zagreb entropy for the graph G = ∼ = ACCN T (β, γ) , is Proof. By using the Table 7 given for the edge varieties of ACCN T (β, γ) graph, third redefined Zagreb index is determined in the Table 8. Now for the entropy of third redefined Zagreb, we will use the the value of third redefined Zagreb index in the Equation 15, and resulted in  = ACN T (m, n) , G 2 = ACSCN T (m, n) and G 3 = ACCN T (m, n) . The plots in the Figures 5 and 6 are extracted from the data in Tables 9 and 10. It shows that the edge weight based VOLUME 4, 2020 first and second Zagreb entropy have different properties for different topologies of nanotubes. For example, the entropy of G 2 shape of nanotube which is ACSCN T (m, n) or semicapped nanotube is growing more rapidly in comparison to other shapes either G 1 and G 3 . Similarly in the Figures 8  and 11 extracted from the Tables 12 and 15, plotting showing the same entropy measure and for semi-capped nanotube, the entropy is growing more rapidly than others. Just in the case shown in the Figures 7 and 10, the plot for entropies are overlapped for capped and semi-capped nanotubes. It shows that each entropy measure is different for different topologies of nanotubes. Either it is single-sided capped or both sided capped. The fact is shown by the exemplary analysis drawn below.