Abstract
Organic compounds such as polyphenylene are very important and useful for the synthesis of many new organic compounds due to their physio-chemical properties. To ascertain these properties, one can use QSPR/QSAR methods which necessitate the computation of topological indices. The topological indices based on two newly introduced abstract notions of ev-degree and ve-degree are in practice to model numerous chemical properties as well as physical properties of organic, inorganic, hybrid, and biological compounds. In this study, we computed a certain number of topological indices for the chemical graph of polyphenylene network which will help to model some of its physio-chemical properties.
1. Introduction
The detailed critical inspection in order to discover essential features or meanings of chemical compounds graphically is known as chemical graph theory. It is the branch of mathematics, which alloys chemistry and graph theory. In graph theory, a simple graph or just graph is constructed by two sets: , the set of vertices, and , the set of edges. Each represents a node in the graph and each denotes the line joining two nodes.
In chemical graph theory, the image obtained from diffraction of X-rays or electron microscopy of a compound (biological or chemical) is drawn into plane and lighted upon its symmetry, and then, peculiarities of this compound is mathematically modeled. The simple sketch of the image of compound is known as the chemical graph where we assume that the ends or vertices are atoms and lines or edges are the bonds between the atoms. Chemical graph theory helps to understand different properties, namely, molecular structure, kinetics of molecules, atoms or electrons, chain or patterns of polymers, crystals and clusters, aromaticity, nuclear magnetic resonance (NMR) analysis, depicting orbitals, and electrons behaviors. Ante Graovac, Alexandru Balaban, Haruo Hosoya, Iv n Gutman, Nenad Trinajstic, and Milan Randic are few scientists who introduced graph theory in chemistry [1].
The job of mathematical modeling the properties of chemical compounds is done by topological indices which we define as a number obtained by a real-valued function,, that is applied to any chemical graph (or molecular structure) of a compound to determine its topology, is known as topological graph index or just topological index (plural: topological indices), whereare edges and vertices of graph, for example, Zagreb indices and their variants, distance indices, detour index, and Wiener index. There are different kinds of topological indices based on degree, distance, and counting [2]. Many physical and chemical properties of different chemical and biological compounds have been modeled mathematically by the aid of topological indices such as boiling point, anti-leishmanial effect, acute toxicity, radial scavenging activity, and many more [3–5]. In this study, we considered some topological indices based on degree of vertices and edges, ve-degree of vertices, and ev-degree of edges. To understand the terms used in formulas of topological indices, first, we consider the following basic definitions [6–8].
For a graph , we have(1)Degree of a vertex means the number of edges connected to the vertex denoted by (2)The number is called degree of edge if is formed by joining the vertex and , denoted by (3)The set is called the open neighborhood of the vertex (4)The set is the closed neighborhood of denoted by (5)The number of nonidentical edges that are incident to each vertex in the closed neighbourhood of the vertex is the ve-degree of vertex denoted by [9](6)If are nodes of an edge , then the order of is equal to the ev-degree [9] of edge , denoted by
For a graph , the topological indices under consideration are given below:(i)Randić index [10]:(ii)ev-degree Randić index [11]:(iii)ve-degree Randić index:(iv)Reciprocal ve-degree Randić index [12]:
Milan Randić, a chemist, introduced the “branching index” in 1975 as a topological index for evaluating the degree of branching in the carbon-atom skeleton of saturated hydrocarbons [10]. Moreover, it is also used to model the cavity surface area of different alcohols. In [11], Suleyman Ediz introduced degree Randić (2) index and proved that it gives more accurate correlation than the previous one. First ve-degree Zagreb beta index: Second ve-degree Zagreb index: Redefined third ve-degree Zagreb index [13]: The modified ev-degree Zagreb index [12]:
Zagreb indices were first used to model anti-inflammatory agility in different acids [14], and then, and were used to model the clearance of cephalosporins and fraction bound in humans [15].(i)ve-degree sum-connectivity index:(ii)ve-degree atom-bond connectivity index: Atom-bond connectivity index is used to model heat of formation in alkanes [16].(iii)ve-degree geometric arithmetic index:(iv)Arithmetic-geometric index [17]:
In [18], the geometrical-arithmetic index (GA) was introduced and used to model the following properties of chemical compounds:(i)Acentric factor (AcenFac)(ii)Boiling point (BP)(iii)Entropy (S)(iv)Enthalpy of formation (HFORM)(v)Enthalpy of vaporization (HVAP)(vi)Standard enthalpy of vaporization (DHVAP)
Recently, some other topological indices (listed below) are introduced on the basis of ev- and ve-degree to model chemical properties written above which give better correlation coefficients of about 0.99088 [19].(i)ve-degree harmonic index:(ii)ev-degree inverse index [12]:(iii)ve-degree inverse sum index [20]:(iv)F-ev-degree index [12]:(v)F-ve-degree index [21]:(vi)First hyper-ve-degree index [21]:(vii)Second hyper-ve-degree index [21]:
2. Polyphenylene Network
Hydrocarbons, which are organic compounds formulated completely by the atoms of hydrogen and carbon elements, are one of the world’s most prominent sources of energy and are found mostly in petroleum form, natural gas, and crude oil. The majority of hydrocarbons are linear chains, rings, or a mix of the two. Polyphenylene, a two-dimensional (2D) limitless hydrocarbon with the formula , that may be formed experimentally, is thermodynamically stable. Furthermore, due to its new structural features, 2D polyphenylene could be used in a variety of industries, such as a barrier for purification. When is produced by the conventional process of steam methane reformation, certain less desirable species are produced, such as , , and . As a result, isolating from these species is critical for its storage and use [25]. SRP is a superb option for many demanding applications including semiconductor components, high-performance bearings, bushings, valves, valve seats, and aircraft substructures, due to its remarkable mechanical, chemical, thermal, and electrical qualities. SRP is an ideal contender for light-weight high-performance applications due to its high specific strength. Antistatic coatings made of electrically conductive polyphenylene (p- or n-doped) are used to protect integrated circuits from static charges, humidity, and corrosion [26]. The following are important performance characteristics:
Mechanical stiffness and strength are extremely high. High compression strength and resistance to pressure, excellent wear and scratch resistance, and good cold temperature stability (to around C), C is a high glass transition temperature. Before and after processing, there is exceptional dimensional stability, low coefficient of thermal expansion (low thermal shrinkage), excellent acid, and basic resistance. With resistance to solvents and hot steam (but lower than PEEK), processability is good (can be extruded and injection molded).
2.1. Mathematical Work and Discussion
In this section, we will compute and discuss all the topological indices mentioned before, for the graph of polyphenylene network , as shown in Figure 1, where, in Figure 2, we show the unit segment of the graph. In the graph, we have vertices joined with single edge, double edges, and triple edges (i.e., ). In the graph of polyphenylene network , we observe that there are vertices and edges. More information about the graph is given in Table 1.
Theorem 1. Letbe the graph of polyphenylene network; then,
Proof. To prove (a), we expand equation (1) and substitute values from Table 1, which givesTo prove (b), we expand (2) and substitute values from Table 1, which givesTo prove (c), we expand (3) and substitute values from Table 1, which givesTo prove (d), we expand (4) and substitute values from Table 1, which gives
Theorem 2. Letbe the graph of polyphenylene network; then,
Proof. To prove (a), expand equation (5) and substitute values from Table 1, which givesTo prove (b), expand (6) and substitute values from Table 1, which givesTo prove (c), expand (7) and substitute values from Table 1, which givesTo prove (d), expand (8) and substitute values from Table 1, which gives
Theorem 3. Let be the graph of polyphenylene network; then,
Proof. To prove (a), expanding equation (9) and substituting values from Table 1, we findTo prove (b), expanding (10) and substituting values from Table 1, we find
Theorem 4. Letbe the graph of polyphenylene network, then
Proof. To prove (a), expand equation (11) and substitute values from Table 1, which givesTo prove (b), expand (12) and substitute values from Table 1, which gives
Theorem 5. Let be the graph of polyphenylene network; then,
Proof. To prove, expand equation (13) and substitute values from Table 1, which gives
Theorem 6. Let be the graph of polyphenylene network; then,
Proof. To prove (a), expand equation (14) and substitute values from Table 1, which givesTo prove (b), expand (15) and substitute values from Table 1, which gives
Theorem 7. Let be the graph of polyphenylene network; then,
Proof. To prove (a), expand equation (16) and substitute values from Table 1, which givesTo prove (b), expand (17) and substitute values from Table 1, which gives
Theorem 8. Let be the graph of polyphenylene network; then,
Proof. To prove (a), expand equation (18) and substitute values from Table 1, which givesTo prove (b), expand (19) and substitute values from Table 1, which gives
2.2. Numerical and Graphical Discussion
In Theorem 1, we stated and then computed Randić indices of the graph of polyphenylene network (Figure 1) for frameworks or units. By the resulting equations, we determined exact numerical values of Randić indices, for some , and listed in Table 2 and plotted them in Figure 3 by which we can observe that all the resulting equations show parabolic behavior. For continuously increasing values of , all the indices go to infinity. increases faster than other indices which are in order .
Table 3 and Figure 4 are derived from Theorem 2 which show the numerical values and graphical pattern of the Zagreb indices of polyphenylene network. The curves are parabolic, and the parabola for has the shortest length of lactusrectum; i.e., it runs much faster than the others.
In Table 4 and Figure 5, we exhibited the numerical measures and graphical style of sum-connectivity and atom-bond connectivity indices from Theorem 3. The curves are parabolic, and the separation between them becomes greater and greater for increasing values of earns larger numerical value than for all values of
For units of polyphenylene, and were given in Theorem 4. By substituting , Table 5 is constructed. In Figure 6, the curves of both the indices are shown from where we can observe that the span between them is growing gradually.
In Figure 7, the outcomes of Theorem 5 are shown. The numerical values are increasing continuously with the increase of frameworks .
Table 6 and Figure 8 are established for Theorem 6 in which we can see that, for all , and both the curves are parabolic in manner. The span between the curves is growing very high.
Numerical values of the indices discussed in Theorem 7 are given in Table 7, where the graph of both the indices is shown in Figure 9. Again, both the curve are parabolic. The index acquires higher values than the index all the time, for the graph of polyphenylene network.
Following are the conclusions of final Theorem 8. The numerical representation is shown in Table 8, and graphical representation is shown in Figure 10. Once again both the curves are parabolic. The separation between them is growing continuously. The numerical values of is always less than the numerical values of .
3. Conclusion
In this study, we calculated 19 topological indices discussed in Section 1 for the graph of polyphenylene network. Each index represents a unique chemical or physical property which is also mentioned in section 1. We also mentioned many qualities of polyphenylene which tell that it is an admirable chemical compound to make new substances and to improve dispositions of the present one. Therefore, the work of this paper will provide assistance in studying the properties of polyphenylene and its derivatives.
Data Availability
The data used to support the findings of this study are cited at relevant places within the article as references.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors equally contributed to this work.