Electron Energy Studying of Molecular Structures via Forgotten Topological Index Computation

1School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China 2Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China 3Department of Mathematics, Riphah Institute of Computing and Applied Sciences, Riphah International University, 14 Ali Road, Lahore, Pakistan 4Department of Applied Mathematics, Iran University of Science and Technology, Tehran 16844, Iran


Introduction
The principal quantum numbers, derived from electrons in an atom or molecule, determine the range of electron energy if we research on the orbital it occupies."Atomic emission," a very common and interesting phenomenon in atoms, proves to be both the origin of the Fraunhofer lines and an effective approach in this field.To illustrate, it describes that once more energy than minimum for the given situation is stored in one electron, it could be emitted as a photon.Fraunhofer lines are another significant technique which turns out to work pretty well in identification and astronomy, where the measurement of "red shift" in stars exactly comes to the point.As we know, spin and angular momentum can help to intensify the whole energy of an electron, so it can be represented by the set of all of its quantum numbers.To illustrate, the speed that an electron orbits is considered as the electron energy in physics and it could be looked upon as the effect of an electron's energy.But some special phenomena still could occur in the real life and even though two electrons get the same speed, they are not exactly the same.For instance, when two electrons are the same except the spin quantum numbers, they can orbit at the same speed.The case is also likely to happen in two electrons only different in angular momentum numbers with one being a positive number and the other being a negative number.
Having computed within the Huckel tight-binding molecular orbital (HMO) approximation, we get   , the total -electron energy.As a quantum-chemical property of conjugated molecules, it turns out to be in good accordance to the thermodynamic properties.Based on the eigenvalues of the adjacency matrix of the molecular graph, we can compute out   , when the conjugated hydrocarbons are still in their ground electronic states.The mentioned graph can be described like this here:   =  + .In the graph,  symbolizes the number of carbon atoms;  is the HMO carbon-atom Coulomband;  is the carbon-carbon resonance integrals; and for the majority (but not all), conjugated -electron systems, 2

Journal of Chemistry
Here,  1 ,  2 , . . .,   represent the eigenvalues of the adjacency matrix A of the underlying molecular graph , forming the spectrum of .
Referring to the relative researches about   and focusing on the research on its dependence on molecular structure in specific, we find that the term  is the only interesting quantity, and it is defined in (1).As a matter of fact, it is a common and traditional way to regard  as the total electron energy, expressed in -units.Hence, there is a need for us to mention that the quantity defined via (1) is called the energy of the graph  in mathematical.
A research on the structure-dependency of total electron energy   in 1972 proposed an approach to the branching of the carbon-atom skeleton by demonstrating that the sum of squares of the vertex degrees of the molecular graph can determine   .What is more, it also pointed that   tends to be influenced by the sum of cubes of degrees of vertices of the molecular graph.Indeed, the formulas for total -electron energy   also concern the sum of cubes of vertex degrees (in many references, this value can be denoted by ∑  3  1 ).In a clear fashion, this quantity is a measure of branching as well.
Thus, in a recent research on the structure-dependency of the total -electron energy, it was indicated that another term on which this energy depends is in the form (see Furtula and Gutman [14]) where (V) is denoted as the degree of vertex V (the number of vertex adjacent to vertex V).In addition, this sum was named forgotten topological index or shortly the -index.
In the web site, http://www.moleculardescriptors.eu/dataset/dataset.htm, the potential ability of the -index was tested using a dataset of octane isomers, which is in accordance to the International Academy of Mathematical Chemistry.In the simplest form, the -index does not recognize heteroatoms and multiple bonds, and this becomes the reason why dataset is chosen as a measure.A list of data including boiling point, melting point, heat capacities, entropy, density, heat of vaporization, enthalpy of formation, motor octane number, molar refraction, acentric factor, total surface area, octanol-water partition coefficient, and molar volume help to compose the octane dataset.The -index shows its strong bonds with most properties here.As a consequence, -index proves to have correlation coefficients greater than 0.95 in entropy and acentric factor.
However, for many other physicochemical characteristics, -index may not be fully correlated.In order to strengthen the predictive ability of -index in potential chemical applications, a linear framework was proposed as follows (see Furtula and Gutman [14]): where  is an adaptive parameter which can be adjusted according to the detailed applications in chemical or pharmacy engineering (generally speaking,  always take value from −20 to 20); the first term ∑ V∈() (() + (V)) was defined as the first Zagreb index which was one of the most traditional indexes in chemical science.By means of a large number of experimental studies, this framework can be used in each of the physicochemical properties with fixed octane database.As an example, an evident improvement can be obtained in the octanol-water partition coefficient, and it was pointed out that the absolute value of the correlation coefficient gets a tight maximum if taking  = −0.14 in the above framework.Then, by virtue of derivation, the octanolwater partition coefficient of octanes can be stated as the following representation: where log  is the logarithm function of the octanol-water partition coefficient.This fact implies that the error of mean absolute percentage is only 0.06% and the correlation coefficient can reach to 0.99896.
In real engineering implements, the model ∑ V∈() (() + (V)) +  ∑ V∈() (() 2 + (V) 2 ) can be regarded as a generalized framework.For different chemical applications, adaptive parameter  is a key factor which can be changed to the optimal value according to the detailed applications.In the above example,  takes −0.14, while for other applications  can take other values and it is determined by detailed physical-chemical properties and measured in the chemical experiments.
In the whole article, the molecular structure is modeled as a graph  with vertex set () and edge set (), where each vertex represents an atom and each edge denotes a chemical bond between two atoms.A topological index defined on the molecular graph can be considered as a real-valued function  :  → R + which maps each chemical structure to a real score.Except the forgotten topological index, there are several famous indices introduced and applied in chemical engineering, such as Wiener index, harmonic index, sum connectivity index, and eccentric index (see Gao et al. [15][16][17][18][19], Yang et al. [20], Marana et al. [21], and Li et al. [22] for more details).The terminologies and notations used but not clearly defined in our paper can be found in Bondy and Mutry [23].
Although there have been many advances in degreebased and distance-based indices of molecular graphs, the studies of forgotten topological index for special chemical molecular structures are still largely limited.For this reason, we give the exact expressions of forgotten topological index for several chemical molecular structures which commonly appeared in various chemical environments.The main trick in our article to deduce the expected result is edge set dividing technology.Let () and Δ() be the minimum and maximum degree of , respectively.The edge set () can be divided into several partitions: for any , 2() ≤  ≤ 2Δ(), let   = { = V ∈ () | (V) + () = }; for any , (()) 2 ≤  ≤ (Δ()) 2 , let  *  = { = V ∈ () | (V)() = }.Using this dividing, the -index can be expressed as More specifically, if we denote   = { = V ∈ () | (V) = , () = }, then the -index can be further stated as In view of this alternation and the detailed analysis of molecular structures, the -index of special chemical graphs can be determined.

Forgotten Topological Index of Nanotubes
In the field of nanomaterial and nanotechnology, there are a large number of new nanostructures being discovered each year.It needs more chemical experiments to figure out their biochemical properties.In this section, we focus on the nanostructures and present the forgotten topological index of some special kinds of nanorelated molecular graphs.

Forgotten Topological Index of TUC 4 [𝑚, 𝑛]
Nanotubes and PAMAM Dendrimers.In the nanoscience, TUC 4 [, ] nanotubes (where  and  are denoted as the number of squares in a row and the number of squares in a column, resp.) are plane tiling of C 4 .This tessellation of C 4 can cover either a torus or a cylinder.The 3D representation of TUC 4 [6, ] is described in Figure 2.
Using the trick of edge set dividing, we yield the following statement.Proof.By observation of PAMAM dendrimer PD 1 , PD 2 , and DS 1 , we ensure that the edge set of PD 1 can be divided into four partitions: The set (PD 2 ) can be divided into four subsets.

Forgotten Topological Index of Two Classes of Polyhex
Nanotubes.As the last part of this section, we aim to study the forgotten topological index of two classes of polyhex nanotubes: zigzag TUZC 6 [, ] and armchair TUAC 6 [, ], where  is the number of hexagons in the first row and  is the number of hexagons in the first column.The molecular structures of TUZC 6 [, ] and TUAC 6 [, ] can be referred to in Figures 3 and 4, respectively.Now, we present the main results in this section.
Index of SC 5 C 7 [, ] Nanotubes and H-Naphtalenic Nanotubes.In nanoscience, SC 5 C 7 [, ] (where  and  express the number of heptagons in each row and the number of periods in whole lattice, resp.)nanotube is a class of C 5 C 7 -net which is yielded by alternating C 5  are denoted as the number of pairs of hexagons in first row and the number of alternative hexagons in a column, resp.) are a trivalent decoration with sequence of C 6 , C 6 , C 4 , C 6 , C 6 , C 4 , . . . in the first row and a sequence of C 6 , C 8 , C 6 , C 8 , . . . in the other rows.In other words, this nanolattice can be considered as a plane tiling of C 4 , C 6 , and C 8 .Therefore, this class of tiling can cover either a cylinder or a torus.Now, our first result on the forgotten topological index of SC 5 C 7 [, ] nanotubes and H-naphtalenic nanotubes is stated as follows.
and C 7 .The standard tiling of C 5 and C 7 can cover either a cylinder or a torus, and each period of SC 5 C 7 [, ] consisted of three rows (more details on th period can be referred to in Figure1).H-Naphtalenic nanotubes NPHX[, ] (where  and