research article electron energy studying of...

Research ArticleElectron Energy Studying of Molecular Structures viaForgotten Topological Index Computation

Wei Gao1 Weifan Wang2 Muhammad Kamran Jamil3 and Mohammad Reza Farahani4

1School of Information Science and Technology Yunnan Normal University Kunming 650500 China2Department of Mathematics Zhejiang Normal University Jinhua 321004 China3Department of Mathematics Riphah Institute of Computing and Applied Sciences Riphah International University14 Ali Road Lahore Pakistan4Department of Applied Mathematics Iran University of Science and Technology Tehran 16844 Iran

Correspondence should be addressed to Wei Gao gaoweiynnueducn

Received 26 April 2016 Revised 9 July 2016 Accepted 26 July 2016

Academic Editor Teodorico C Ramalho

Copyright copy 2016 Wei Gao et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

It is found from the earlier studies that the structure-dependency of total 120587-electron energy 119864120587 heavily relies on the sum of squaresof the vertex degrees of the molecular graph Hence it provides a measure of the branching of the carbon-atom skeleton In recentyears the sum of squares of the vertex degrees of themolecular graph has been defined as forgotten topological index which reflectsthe structure-dependency of total 120587-electron energy 119864120587 and measures the physical-chemical properties of molecular structures Inthis paper in order to research the structure-dependency of total 120587-electron energy 119864120587 we present the forgotten topological indexof some important molecular structures from mathematical standpoint The formulations we obtained here use the approach ofedge set dividing and the conclusions can be applied in physics chemical material and pharmaceutical engineering

1 Introduction

The principal quantum numbers derived from electrons inan atom or molecule determine the range of electron energyif we research on the orbital it occupies ldquoAtomic emissionrdquo avery common and interesting phenomenon in atoms provesto be both the origin of the Fraunhofer lines and an effectiveapproach in this field To illustrate it describes that oncemoreenergy than minimum for the given situation is stored in oneelectron it could be emitted as a photon Fraunhofer lines areanother significant technique which turns out to work prettywell in identification and astronomy where themeasurementof ldquored shiftrdquo in stars exactly comes to the point As weknow spin and angular momentum can help to intensify thewhole energy of an electron so it can be represented by theset of all of its quantum numbers To illustrate the speedthat an electron orbits is considered as the electron energyin physics and it could be looked upon as the effect of anelectronrsquos energy But some special phenomena still couldoccur in the real life and even though two electrons get thesame speed they are not exactly the same For instance when

two electrons are the same except the spin quantumnumbersthey can orbit at the same speed The case is also likely tohappen in two electrons only different in angular momentumnumbers with one being a positive number and the otherbeing a negative number

Having computed within the Huckel tight-bindingmolecular orbital (HMO) approximation we get 119864120587 thetotal 120587-electron energy As a quantum-chemical property ofconjugated molecules it turns out to be in good accordanceto the thermodynamic properties Based on the eigenvaluesof the adjacency matrix of the molecular graph we cancompute out 119864120587 when the conjugated hydrocarbons arestill in their ground electronic states The mentioned graphcan be described like this here 119864120587 = 119899120572 + 119864120573 In thegraph 119899 symbolizes the number of carbon atoms 120572 is theHMO carbon-atom Coulomband 120573 is the carbon-carbonresonance integrals and for the majority (but not all)conjugated 120587-electron systems

119864 = 119899sum119894=1

10038161003816100381610038161205821198941003816100381610038161003816 (1)

Hindawi Publishing CorporationJournal of ChemistryVolume 2016 Article ID 1053183 7 pageshttpdxdoiorg10115520161053183

2 Journal of Chemistry

Here 1205821 1205822 120582119899 represent the eigenvalues of the adja-cencymatrixA of the underlyingmolecular graph119866 formingthe spectrum of 119866

Referring to the relative researches about119864120587 and focusingon the research on its dependence on molecular structurein specific we find that the term 119864 is the only interestingquantity and it is defined in (1) As a matter of fact it isa common and traditional way to regard 119864 as the total 120587-electron energy expressed in 120573-units Hence there is a needfor us to mention that the quantity defined via (1) is called theenergy of the graph 119866 in mathematical

More details on total 120587-electron energy can be found inGutman et al [1ndash5] Angelina et al [6] Turker and Gutman[7] Jones et al [8] Radenkovic and Gutman [9] Peric et al[10] Morales [11] Markovic [12] and Morales [13]

A research on the structure-dependency of total 120587-electron energy 119864120587 in 1972 proposed an approach to the bra-nching of the carbon-atom skeleton by demonstrating thatthe sum of squares of the vertex degrees of the moleculargraph can determine 119864120587 What is more it also pointed that119864120587 tends to be influenced by the sum of cubes of degreesof vertices of the molecular graph Indeed the formulas fortotal 120587-electron energy 119864120587 also concern the sum of cubes ofvertex degrees (in many references this value can be denotedby sum12059031) In a clear fashion this quantity is a measure ofbranching as well

Thus in a recent research on the structure-dependency ofthe total 120587-electron energy it was indicated that another termon which this energy depends is in the form (see Furtula andGutman [14])

119865 (119866) = sumVisin119881(119866)

119889 (V)3 = sum119906Visin119864(119866)

(119889 (119906)2 + 119889 (V)2) (2)

where 119889(V) is denoted as the degree of vertex V (the number ofvertex adjacent to vertex V) In addition this sum was namedforgotten topological index or shortly the 119865-index

In theweb site httpwwwmoleculardescriptorseudata-setdatasethtm the potential ability of the119865-index was testedusing a dataset of octane isomers which is in accordanceto the International Academy of Mathematical ChemistryIn the simplest form the 119865-index does not recognize het-eroatoms and multiple bonds and this becomes the reasonwhy dataset is chosen as a measure A list of data includingboiling point melting point heat capacities entropy densityheat of vaporization enthalpy of formation motor octanenumber molar refraction acentric factor total surface areaoctanol-water partition coefficient and molar volume helpto compose the octane dataset The 119865-index shows its strongbonds with most properties here As a consequence 119865-indexproves to have correlation coefficients greater than 095 inentropy and acentric factor

However for many other physicochemical character-istics 119865-index may not be fully correlated In order tostrengthen the predictive ability of 119865-index in potentialchemical applications a linear framework was proposed asfollows (see Furtula and Gutman [14])

sum119906Visin119864(119866)

(119889 (119906) + 119889 (V)) + 120582 sum119906Visin119864(119866)

(119889 (119906)2 + 119889 (V)2) (3)

where 120582 is an adaptive parameter which can be adjustedaccording to the detailed applications in chemical or phar-macy engineering (generally speaking 120582 always take valuefrom minus20 to 20) the first term sum119906Visin119864(119866)(119889(119906) + 119889(V)) wasdefined as the first Zagreb index which was one of the mosttraditional indexes in chemical science By means of a largenumber of experimental studies this framework can be usedin each of the physicochemical properties with fixed octanedatabase As an example an evident improvement can beobtained in the octanol-water partition coefficient and itwas pointed out that the absolute value of the correlationcoefficient gets a tight maximum if taking 120582 = minus014 in theabove framework Then by virtue of derivation the octanol-water partition coefficient of octanes can be stated as thefollowing representation

log119875 = minus02058( sum119906Visin119864(119866)

(119889 (119906) + 119889 (V))

minus 014 sum119906Visin119864(119866)

(119889 (119906)2 + 119889 (V)2)) + 75864(4)

where log119875 is the logarithm function of the octanol-waterpartition coefficient This fact implies that the error ofmean absolute percentage is only 006 and the correlationcoefficient can reach to 099896

In real engineering implements themodelsum119906Visin119864(119866)(119889(119906)+119889(V)) + 120582sum119906Visin119864(119866)(119889(119906)2 + 119889(V)2) can be regarded as ageneralized framework For different chemical applicationsadaptive parameter 120582 is a key factor which can be changed tothe optimal value according to the detailed applications Inthe above example 120582 takes minus014 while for other applications120582 can take other values and it is determined by detailedphysical-chemical properties and measured in the chemicalexperiments

In thewhole article themolecular structure ismodeled asa graph119866with vertex set119881(119866) and edge set 119864(119866) where eachvertex represents an atom and each edge denotes a chemicalbond between two atoms A topological index defined on themolecular graph can be considered as a real-valued function119891 119866 rarr R+ which maps each chemical structure to areal score Except the forgotten topological index there areseveral famous indices introduced and applied in chemicalengineering such as Wiener index harmonic index sumconnectivity index and eccentric index (see Gao et al [15ndash19] Yang et al [20] Marana et al [21] and Li et al [22] formore details) The terminologies and notations used but notclearly defined in our paper can be found in Bondy andMutry[23]

Although there have been many advances in degree-based and distance-based indices of molecular graphs thestudies of forgotten topological index for special chemicalmolecular structures are still largely limited For this reasonwe give the exact expressions of forgotten topological indexfor several chemical molecular structures which commonlyappeared in various chemical environments

Journal of Chemistry 3

3m3m minus 1

3m minus 2

Figure 1 119894th period of SC5C7[119901 119902] nanotube

The rest of the context is arranged as follows firstwe present the forgotten topological index of some nanos-tructures SC5C7[119901 119902] nanotubes H-naphtalenic nanotubesNPHX[119898 119899] TUC4[119898 119899] nanotubes PAMAM dendrimersPD1[119899] PD2[119899] and DS1[119899] VC5C7[119901 119902] and HC5C7[119901 119902]nanotubes zigzag TUZC6[119898 119899] and armchair TUAC6[119898 119899]nanotubes second we calculate the forgotten topologicalindex of several kinds of polyomino chains

Themain trick in our article to deduce the expected resultis edge set dividing technology Let 120575(119866) and Δ(119866) be theminimum and maximum degree of 119866 respectively The edgeset 119864(119866) can be divided into several partitions for any 1198942120575(119866) le 119894 le 2Δ(119866) let 119864119894 = 119890 = 119906V isin 119864(119866) | 119889(V) + 119889(119906) = 119894for any 119895 (120575(119866))2 le 119895 le (Δ(119866))2 let 119864lowast119895 = 119890 = 119906V isin119864(119866) | 119889(V)119889(119906) = 119895 Using this dividing the 119865-index canbe expressed as

119865 (119866) = sum119864119894

sum119906Visin119864119894

(119889 (119906)2 + 119889 (V)2)= sum119864lowast119895

sum119906Visin119864lowast119895

(119889 (119906)2 + 119889 (V)2) (5)

More specifically if we denote 119864119894119895 = 119890 = 119906V isin 119864(119866) | 119889(V) =119894 119889(119906) = 119895 then the 119865-index can be further stated as

119865 (119866) = sum119864119894119895

1003816100381610038161003816100381611986411989411989510038161003816100381610038161003816 (1198942 + 1198952) (6)

In view of this alternation and the detailed analysis ofmolecular structures the 119865-index of special chemical graphscan be determined

2 Forgotten Topological Index of Nanotubes

In the field of nanomaterial and nanotechnology there are alarge number of new nanostructures being discovered eachyear It needs more chemical experiments to figure out theirbiochemical properties In this section we focus on thenanostructures and present the forgotten topological indexof some special kinds of nanorelated molecular graphs

21 Forgotten Topological Index of SC5C7[119901 119902] Nanotubesand H-Naphtalenic Nanotubes In nanoscience SC5C7[119901 119902](where 119901 and 119902 express the number of heptagons in each rowand the number of periods in whole lattice resp) nanotubeis a class of C5C7-net which is yielded by alternating C5and C7 The standard tiling of C5 and C7 can cover either acylinder or a torus and each period of SC5C7[119901 119902] consistedof three rows (more details on 119894th period can be referred to inFigure 1)

H-Naphtalenic nanotubes NPHX[119898 119899] (where 119898 and119899 are denoted as the number of pairs of hexagons in

first row and the number of alternative hexagons in acolumn resp) are a trivalent decoration with sequence ofC6C6C4C6C6C4 in the first row and a sequence ofC6C8C6C8 in the other rows In other words thisnanolattice can be considered as a plane tiling of C4 C6 andC8Therefore this class of tiling can cover either a cylinder ora torus

Now our first result on the forgotten topological indexof SC5C7[119901 119902] nanotubes and H-naphtalenic nanotubes isstated as follows

Theorem 1 One has

119865 (SC5C7 [119901 119902]) = 216119901119902 minus 76119901119865 (NPHX [119898 119899]) = 270119898119899 minus 76119898 (7)

Proof According to the molecular graph structureSC5C7[119901 119902] we see that |119864(SC5C7[119901 119902])| = 12119901119902 minus 2119901and its edge set can be separated into three subsets

(i) 1198644 (or 119864lowast4 ) 119889(119906) = 119889(V) = 2 |1198644| = |119864lowast4 | = 119901(ii) 1198646 (or 119864lowast9 ) 119889(119906) = 119889(V) = 3 |1198646| = |119864lowast9 | = 12119901119902minus9119901(iii) 1198645 (or 119864lowast6 ) 119889(119906) = 2 119889(V) = 3 and |1198645| = |119864lowast6 | = 6119901ForH-naphtalenic nanotubesNPHX[119898 119899] we check that|119864(NPHX[119898 119899])| = 15119898119899minus2119898 and its edge set can be divided

into two partitions

(i) 1198646 (or 119864lowast9 ) 119889(119906) = 119889(V) = 3 |1198646| = |119864lowast9 | = 15119898119899 minus10119898(ii) 1198645 (or 119864lowast6 ) 119889(119906) = 2 119889(V) = 3 |1198645| = |119864lowast6 | = 8119898

At last we get the desired formulations in terms of thedefinitions of forgotten topological index

22 Forgotten Topological Index of TUC4[119898 119899] Nanotubesand PAMAM Dendrimers In the nanoscience TUC4[119898 119899]nanotubes (where 119898 and 119899 are denoted as the number ofsquares in a row and the number of squares in a column resp)are plane tiling of C4 This tessellation of C4 can cover eithera torus or a cylinder The 3D representation of TUC4[6 119899] isdescribed in Figure 2

Using the trick of edge set dividing we yield the followingstatement

Theorem 2 One has

119865 (TUC4 [119898 119899]) = 64119898119899 minus 10119898 (8)

Proof By observing the structure of TUC4[119898 119899] we verifythat |119864(TUC4[119898 119899])| = 2119898119899 + 119898 and its edge set can bedivided into three partitions

(i) 1198646 (or 119864lowast9 ) 119889(119906) = 119889(V) = 3 |1198646| = |119864lowast9 | = 2119898(ii) 1198647 (or 119864lowast12) 119889(119906) = 3 119889(V) = 4 |1198647| = |119864lowast12| = 2119898(iii) 1198648 (or 119864lowast16)119889(119906) = 119889(V) = 4 |1198648| = |119864lowast16| = 119898(2119899minus3)Therefore the expected formulation is followed by the

definitions of forgotten topological index


Figure 2 The 3D expression of TUC4[6 119899]

Now we use PD1 to denote PAMAMdendrimers with tri-functional core unit generated by dendrimer generations 119866119899with 119899 growth stages use PD2 to denote PAMAM dendrimerwith different core generated by dendrimer generations 119866119899with 119899 growth stages and use DS1 to express other kinds ofPAMAM dendrimer with 119899 growth stages

Theorem 3 One has

119865 (PD1) = 492 sdot 2119899 minus 258119865 (PD2) = 656 sdot 2119899 minus 310119865 (DS1) = 228 sdot 3119899 minus 160

(9)

Proof By observation of PAMAM dendrimer PD1 PD2 andDS1 we ensure that the edge set of PD1 can be divided intofour partitions

(i) 1198643 (or 119864lowast2 ) 119889(119906) = 1 119889(V) = 2 |1198643| = |119864lowast2 | = 3 sdot 2119899(ii) 119864lowast3 119889(119906) = 1 119889(V) = 3 |119864lowast3 | = 6 sdot 2119899 minus 3(iii) 119864lowast4 119889(119906) = 119889(V) = 2 |119864lowast4 | = 18 sdot 2119899 minus 9(iv) 1198645 (or 119864lowast6 ) 119889(119906) = 2 119889(V) = 3 |1198645| = |119864lowast6 | = 21 sdot 2119899 minus12The set 119864(PD2) can be divided into four subsets

(i) 1198643 (or 119864lowast2 ) 119889(119906) = 1 119889(V) = 2 |1198643| = |119864lowast2 | = 4 sdot 2119899(ii) 119864lowast3 119889(119906) = 1 119889(V) = 3 |119864lowast3 | = 8 sdot 2119899 minus 4(iii) 119864lowast4 119889(119906) = 119889(V) = 2 |119864lowast4 | = 24 sdot 2119899 minus 11(iv) 1198645 (or 119864lowast6 ) 119889(119906) = 2 119889(V) = 3 |1198645| = |119864lowast6 | = 28 sdot 2119899 minus14

Figure 3 The 3D lattice of the zigzag TUZC6[10 7]

Moreover the edge set of DS1 can be divided into threeparts

(i) 1198645 119889(119906) = 1 119889(V) = 4 |1198645| = 4 sdot 3119899(ii) 1198644 119889(119906) = 119889(V) = 2 |1198644| = 10 sdot 3119899 minus 10(iii) 1198646 (or 119864lowast8 ) 119889(119906) = 2 119889(V) = 4 |1198646| = |119864lowast8 | = 4 sdot 3119899 minus4Finally the results are deduced bymeans of the definitions

of forgotten topological index

23 Forgotten Topological Index of VC5C 7[119901 119902] andHC5C 7[119901 119902] Nanotubes The aim of this subsection isto compute the forgotten topological index of VC5C7[119901 119902]and HC5C7[119901 119902] (where 119901 is the number of pentagons in thefirst row and 119902 is the number of repetitions in four first rowsof vertices and edges) nanotubes The molecular structuresof these nanotubes consist of cycles C5 and C7 (C5C7 netwhich is a trivalent decoration constructed by alternating C5and C7) by different compound It can cover either a cylinderor a torus

Now we present the main results in this subsection

Theorem 4 One has

119865 (VC5C7 [119901 119902]) = 432119901119902 + 48119901119865 (HC5C7 [119901 119902]) = 216119901119902 + 40119901 (10)

Proof By graph analysis and computation we have|119864(VC5C7[119901 119902])| = 24119901119902 + 6119901 and |119864(HC5C7[119901 119902])| =12119901119902 + 5119901 Furthermore we yield two partitions of edge set119864(VC5C7[119901 119902]) 1198646 119889(119906) = 119889(V) = 3 1198645 119889(119906) = 2 and119889(V) = 3 and three partitions of edge set 119864(HC5C7[119901 119902]) 1198644 119889(119906) = 119889(V) = 2 1198646 119889(119906) = 119889(V) = 3 1198645 119889(119906) = 2 and119889(V) = 3Consider nanotubes VC5C7[119901 119902] with any 119901 119902 isin N we

get |1198645| = |119864lowast6 | = 12119901 and |1198646| = |119864lowast9 | = 24119901119902 minus 6119901 Fornanotube HC5C7[119901 119902] with any 119901 119902 isin N we deduce |1198644| =|119864lowast4 | = 119901 |1198645| = |119864lowast6 | = 8119901 and |1198646| = |119864lowast9 | = 12119901119902 minus 4119901

Therefore in terms of definitions of the forgotten topo-logical index we get the final conclusion as expected

24 Forgotten Topological Index of Two Classes of PolyhexNanotubes As the last part of this section we aim to studythe forgotten topological index of two classes of polyhexnanotubes zigzag TUZC6[119898 119899] and armchair TUAC6[119898 119899]where 119898 is the number of hexagons in the first row and 119899 isthe number of hexagons in the first column The molecularstructures of TUZC6[119898 119899] and TUAC6[119898 119899] can be referredto in Figures 3 and 4 respectively

Now we present the main results in this section


12

3

2

3

m

n

middot middot middot


middotmiddotmiddot

middotmiddotmiddot

m minus 1

n minus 1

Figure 4 The 3D lattice of the armchair TUAC6[119898 119899]

Theorem 5 One has

119865 (TUZC6 [119898 119899]) = 54119898119899 + 16119898119865 (TUAC6 [119898 119899]) = 54119898119899 + 16119898 (11)

Proof It is not hard to check that |119864(TUZC6[119898 119899])| =|119864(TUAC6[119898 119899])| = 3119898119899 + 2119898 Consider nanotubesTUZC6[119898 119899] with any 119898 119899 isin N we deduce that its edge setcan be divided into two parts

(i) 1198646 (or 119864lowast9 ) 119889(119906) = 119889(V) = 3 and |1198646| = |119864lowast9 | = 3119898119899 minus2119898(ii) 1198645 (or 119864lowast6 ) 119889(119906) = 2 119889(V) = 3 and |1198645| = |119864lowast6 | = 4119898

For nanotubes TUAC6[119898 119899] with any 119898 119899 isin N we find thatits edge set can be divided into three parts

(i) 1198644 (or 119864lowast4 ) 119889(119906) = 119889(V) = 2 and |1198644| = |119864lowast4 | = 119898(ii) 1198646 (or 119864lowast9 ) 119889(119906) = 119889(V) = 3 and |1198646| = |119864lowast9 | = 3119898119899minus119898(iii) 1198645 (or 119864lowast6 ) 119889(119906) = 2 119889(V) = 3 and |1198645| = |119864lowast6 | =2119898

Therefore the conclusion is deduced in view of thedefinition of forgotten topological index

3 Forgotten Topological Index ofPolyomino Chains

From the geometric point of view a polyomino system is afinite 2-connected plane graph in which each interior cellis encircled by a regular square In other words it is anedge-connected union of cells in the planar square latticePolyomino chain is a particular polyomino system such thatthe joining of the centers (set 119888119894 as the center of the 119894th square)of its adjacent regular composes a path 1198881 1198882 119888119899 Let B119899 bethe set of polyomino chains with 119899 squares There is 3119899 + 1edges in every 119861119899 isin B119899 where 119861119899 is named as a linear chainand denoted by 119871119899 if the subgraph of 119861119899 induced by thevertices with 119889(V) = 3 is a molecular graph with exactly 119899 minus 2squares Also 119861119899 can be called a zigzag chain and labelled

as 119885119899 if the subgraph of 119861119899 is induced by the vertices with119889(V) gt 2 is 119875119899The angularly connected squares or branched is a kink

of a polyomino chain A maximal linear chain (containingthe terminal squares and kinks at its end) in the polyominochains is called a segment of polyomino chain Let 119897(119878) be thelength of 119878 which is calculated by the number of squares in119878 For any segment 119878 of a polyomino chain we get 119897(119878) isin[2 119899] Furthermore we deduce 1198971 = 119899 and 119898 = 1 for alinear chain 119871119899 with 119899 squares and 119897119894 = 2 and119898 = 119899 minus 1 for azigzag chain 119885119899 with 119899 squares

Inwhat follows we always assume that a polyomino chainconsists of a sequence of segments 1198781 1198782 119878119898 and 119897(119878119894) = 119897119894where 119898 ge 1 and 119894 isin 1 2 119898 We derive that sum119898119894=1 119897119894 =119899 + 119898 minus 1

The theorems presented in the following can deducethe expression of forgotten topological index of polyominochains

Theorem 6 Let 119871119899 119885119899 be the polyomino chains explainedabove One gets

119865 (119871119899) = 54119899 minus 22119865 (119885119899) =

32 119899 = 172119899 minus 58 119899 ge 2

(12)

Proof For 119899 = 1 we can check the results directly In thefollowing consideration we always suppose that 119899 ge 2

It is not hard to verify that |119864(119871119899)| = |119864(119885119899)| = 3119899+1 Let119899119894119895 = 1003816100381610038161003816(119906 V) | 119889 (119906) = 119894 119889 (V) = 1198951003816100381610038161003816 (13)

(i) For the polyomino chain 119871119899 we have 11989922 = 2 11989923 = 4and 11989933 = 3119899 minus 5

(ii) For zigzag chain 119885119899 we obtain 11989922 = 11989934 = 2 11989923 = 411989924 = 2(119898 minus 1) and 11989944 = 3119899 minus 2119898 minus 5Using 119898 = 119899 minus 1 for 119885119899 and the definition of forgotten

topological index we yield the desired results

Theorem 7 Let 1198611119899 (119899 ge 3) be a polyomino chain with 119899squares and of 119898 segments 1198781 1198782 satisfy 1198971 = 2 and 1198972 = 119899 minus 1Then one has

119865 (1198611119899) = 54119899 minus 4 (14)

Proof It is trivial for 119899 = 3 we omit the detailed proof hereFor 119899 ge 4 we get 11989922 = 2 11989923 = 5 11989924 = 1 11989934 = 3 and 11989933 =3119899 minus 10 Therefore by virtue of the definition of forgottentopological index the desired result is obtained

We assume that 2 le 119897(119894) le 119899 minus 1 with 1 le 119894 le 119898 in thefollowing consideration of this section

Theorem 8 Let 1198612119899 (119899 ge 4) be a polyomino chain with 119899squares and119898 segments 1198781 1198782 119878119898 (119898 ge 3) satisfy 1198971 = 119897119898 =2 and 1198972 119897119898minus1 ge 3 Then one infers

119865 (1198612119899) = 54119899 + 18119898 minus 40 (15)


Proof For 1198612119899 with 119899 ge 4 we yield 11989922 = 2 11989923 = 2119898 11989924 = 211989934 = 4119898 minus 6 and 11989933 = 3119899 minus 6119898 + 3 Hence according tothe definition of forgotten topological index we derive thedesired results

In a similar way we deduce the following last twoconclusions in our paper

Theorem 9 Let 1198613119899 be a polyomino chain with 119899 squares and119898 segments 1198781 1198782 119878119898 (119898 ge 3) satisfy 1198971 = 2 1198972 119897119898minus1119897119898 ge 3 or 119897119898 = 2 1198971 1198972 119897119898minus1 ge 3 Then one obtains

119865 (1198613119899) = 18119898 + 54119899 minus 40 (16)

Theorem 10 Let 1198614119899 be a polyomino chain with 119899 squaresand 119898 segments 1198781 1198782 119878119898 (119898 ge 3) meet 119897119894 ge 3 (119894 isin1 2 119898) Then one has

119865 (1198614119899) = 18119898 + 54119899 minus 40 (17)

Remark 11 After the manuscript was in press we received acomment from Akbar Ali It pointed thatTheorems 6ndash10 canbe obtained fromTheorem 31 in Ali et al [24] By comparingTheorem 31 and our results Theorems 6ndash10 we found thatthe results presented in our paper set the different parameterfrom the result in Ali et al [24] More importantly the tricksused in our paper are completely different from the formerone All the results yielded in our paper in light of edge setdividing which raised the edge classification in detail Fromthis point of view the conclusions Theorems 6ndash10 are stillvaluable to the readers

4 Conclusion

In this paper by means of analyzing of molecular graphstructure edge set dividing approach and mathematicalderivation we report the forgotten topological index ofcertain important and widely appeared chemical structuressuch as SC5C7[119901 119902] nanotubes H-naphtalenic nanotubesNPHX[119898 119899] TUC4[119898 119899] nanotubes and some classes ofpolyomino chains These theoretical results achieved in ourpaper reveal the structure-dependency characteristic of total120587-electron energy119864120587 for thesemolecular structures and illus-trate the promising prospects of chemical pharmaceuticalsandmaterials engineering applications such as octanol-waterpartition coefficient measure and the test of melting pointboiling point

Regrettably due to the lack of equipment and other exper-imental conditions we did not present a specific chemicalexperiment to show how to apply the theoretical results tospecific chemical engineering whereas we believe the resultsobtained in this paper can provide theoretical support for thechemical pharmaceutical medicine and materials scienceresearch

Finally we find out the following problems which maybecome the further topics for the studies in this field

(i) What are the upper and lower bound of119865-index in thesetting that some graph parameters (diameter vertexconnectivity or edge connectivity chromatic number

etc) are fixed And which molecular structures canreach these maximum or minimum numbers

(ii) In many engineering applications in chemical mate-rial medicine and pharmaceutical field how is 119865-index used in these fields to test the physical chemi-cal biology and pharmacological properties

(iii) What is the relationship between 119865-index and otherexisting degree-based indices (such as atom-bondconnectivity index first and second Zagreb indicesRandic index Balaban index Narumi Katayamaindex geometric-arithmetic index sum connectivityindex harmonic index etc)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China (nos 11401519 and 61262070)The authors earnestly appreciate all Akbar Alirsquos warm helpearnestly for providing a valuable comment and correctingsome mistakes on the computation results

References

[1] I Gutman ldquoTotal 120587-electron energy of conjugated moleculeswith non-bonding molecular orbitalsrdquo Zeitschrift fur Natur-forschungmdashSection A Journal of Physical Sciences vol 71 no 2pp 161ndash164 2016

[2] I Gutman G Indulal and R Todeschini ldquoGeneralizing theMcClelland bounds for total 120587-electron energyrdquo Zeitschrift furNaturforschung A vol 63 no 5-6 pp 280ndash282 2008

[3] I Gutman S Gojak B Furtula S Radenkovic and AVodopivec ldquoRelating total 120587-electron energy and resonanceenergy of benzenoid molecules with Kekule- and Clar-structure-based parametersrdquo Monatshefte fur Chemie vol 137no 9 pp 1127ndash1138 2006

[4] I Gutman N Cmiljanovic S Milosavljevic and S RadenkovicldquoEffect of non-bonding molecular orbitals on total 120587-electronenergyrdquo Chemical Physics Letters vol 383 no 1-2 pp 171ndash1752004

[5] I Gutman and K C Das ldquoEstimating the total 120587-electronenergyrdquo Journal of the Serbian Chemical Society vol 78 no 12pp 1925ndash1933 2013

[6] E L Angelina D J R Duarte and N M Peruchena ldquoIsthe decrease of the total electron energy density a covalenceindicator in hydrogen and halogen bondsrdquo Journal ofMolecularModeling vol 19 no 5 pp 2097ndash2106 2013

[7] L Turker and I Gutman ldquoIterative estimation of total 120587-electron energyrdquo Journal of the Serbian Chemical Society vol70 no 10 pp 1193ndash1197 2005

[8] N C Jones D Field and J-P Ziesel ldquoLow-energy total electronscattering in the methyl halides CH3Cl CH3Br and CH3IrdquoInternational Journal of Mass Spectrometry vol 277 no 1ndash3 pp91ndash95 2008


[9] S Radenkovic and I Gutman ldquoTotal 120587-electron energy andLaplacian energy how far the analogy goesrdquo Journal of theSerbian Chemical Society vol 72 no 12 pp 1343ndash1350 2007

[10] M Peric I Gutman and J Radic-Peric ldquoThe Huckel total 120587-electron energy puzzlerdquo Journal of the Serbian Chemical Societyvol 71 no 7 pp 771ndash783 2006

[11] D A Morales ldquoThe total 120587-electron energy as a problem ofmoments application of the Backus-Gilbert methodrdquo Journalof Mathematical Chemistry vol 38 no 3 pp 389ndash397 2005

[12] S Markovic ldquoApproximating total 120587-electron energy ofphenylenes in terms of spectral momentsrdquo Indian Journalof Chemistry Section AmdashInorganic Bio-Inorganic PhysicalTheoretical amp Analytical Chemistry vol 42 no 6 pp 1304ndash1308 2003

[13] D AMorales ldquoSystematic search of bounds for total 120587-electronenergyrdquo International Journal of Quantum Chemistry vol 93no 1 pp 20ndash31 2003

[14] B Furtula and I Gutman ldquoA forgotten topological indexrdquoJournal of Mathematical Chemistry vol 53 no 4 pp 1184ndash11902015

[15] W Gao and M R Farahani ldquoComputing the reverse eccentricconnectivity index for certain family of nanocone and fullerenestructuresrdquo Journal of Nanotechnology vol 2016 Article ID3129561 6 pages 2016

[16] W Gao andW FWang ldquoSecond atom-bond connectivity indexof special chemical molecular structuresrdquo Journal of Chemistryvol 2014 Article ID 906254 8 pages 2014

[17] W Gao and W Wang ldquoThe vertex version of weighted Wienernumber for bicyclic molecular structuresrdquo Computational andMathematical Methods inMedicine vol 2015 Article ID 41810610 pages 2015

[18] W Gao W F Wang and M R Farahani ldquoTopological indicesstudy of molecular structure in anticancer drugsrdquo Journal ofChemistry vol 2016 Article ID 3216327 8 pages 2016

[19] WGao andMR Farahani ldquoDegree-based indices computationfor special chemical molecular structures using edge dividingmethodrdquoAppliedMathematics andNonlinear Sciences vol 1 no1 pp 99ndash122 2016

[20] F Yang X Wang M Li et al ldquoTemplated synthesis of single-walled carbon nanotubes with specific structurerdquo Accounts ofChemical Research vol 49 no 4 pp 606ndash615 2016

[21] N L Marana A R Albuquerque F A La Porta E Longo andJ R Sambrano ldquoPeriodic density functional theory study ofstructural and electronic properties of single-walled zinc oxideand carbonnanotubesrdquo Journal of Solid State Chemistry vol 237pp 36ndash47 2016

[22] S N Li J B Liu and B X Liu ldquoFirst principles study ofnanostructured TiS2 electrodes for Na and Mg ion storagerdquoJournal of Power Sources vol 320 pp 322ndash331 2016

[23] J A Bondy and U S R Mutry Graph Theory Springer BerlinGermany 2008

[24] A Ali A A Bhatti and Z Raza ldquoA note on the zeroth-ordergeneral Randic index of cacti and polyomino chainsrdquo IranianJournal of Mathematical Chemistry vol 5 pp 143ndash152 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy


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Chemistry


Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of


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Applied ChemistryJournal of



Theoretical ChemistryJournal of


Journal of

Spectroscopy

Analytical ChemistryInternational Journal of


Journal of


Quantum Chemistry


Organic Chemistry International

ElectrochemistryInternational Journal of



CatalystsJournal of


Here 1205821 1205822 120582119899 represent the eigenvalues of the adja-cencymatrixA of the underlyingmolecular graph119866 formingthe spectrum of 119866

Referring to the relative researches about119864120587 and focusingon the research on its dependence on molecular structurein specific we find that the term 119864 is the only interestingquantity and it is defined in (1) As a matter of fact it isa common and traditional way to regard 119864 as the total 120587-electron energy expressed in 120573-units Hence there is a needfor us to mention that the quantity defined via (1) is called theenergy of the graph 119866 in mathematical

More details on total 120587-electron energy can be found inGutman et al [1ndash5] Angelina et al [6] Turker and Gutman[7] Jones et al [8] Radenkovic and Gutman [9] Peric et al[10] Morales [11] Markovic [12] and Morales [13]

A research on the structure-dependency of total 120587-electron energy 119864120587 in 1972 proposed an approach to the bra-nching of the carbon-atom skeleton by demonstrating thatthe sum of squares of the vertex degrees of the moleculargraph can determine 119864120587 What is more it also pointed that119864120587 tends to be influenced by the sum of cubes of degreesof vertices of the molecular graph Indeed the formulas fortotal 120587-electron energy 119864120587 also concern the sum of cubes ofvertex degrees (in many references this value can be denotedby sum12059031) In a clear fashion this quantity is a measure ofbranching as well

Thus in a recent research on the structure-dependency ofthe total 120587-electron energy it was indicated that another termon which this energy depends is in the form (see Furtula andGutman [14])

119865 (119866) = sumVisin119881(119866)

119889 (V)3 = sum119906Visin119864(119866)

(119889 (119906)2 + 119889 (V)2) (2)

where 119889(V) is denoted as the degree of vertex V (the number ofvertex adjacent to vertex V) In addition this sum was namedforgotten topological index or shortly the 119865-index

In theweb site httpwwwmoleculardescriptorseudata-setdatasethtm the potential ability of the119865-index was testedusing a dataset of octane isomers which is in accordanceto the International Academy of Mathematical ChemistryIn the simplest form the 119865-index does not recognize het-eroatoms and multiple bonds and this becomes the reasonwhy dataset is chosen as a measure A list of data includingboiling point melting point heat capacities entropy densityheat of vaporization enthalpy of formation motor octanenumber molar refraction acentric factor total surface areaoctanol-water partition coefficient and molar volume helpto compose the octane dataset The 119865-index shows its strongbonds with most properties here As a consequence 119865-indexproves to have correlation coefficients greater than 095 inentropy and acentric factor

However for many other physicochemical character-istics 119865-index may not be fully correlated In order tostrengthen the predictive ability of 119865-index in potentialchemical applications a linear framework was proposed asfollows (see Furtula and Gutman [14])

sum119906Visin119864(119866)

(119889 (119906) + 119889 (V)) + 120582 sum119906Visin119864(119866)

(119889 (119906)2 + 119889 (V)2) (3)

where 120582 is an adaptive parameter which can be adjustedaccording to the detailed applications in chemical or phar-macy engineering (generally speaking 120582 always take valuefrom minus20 to 20) the first term sum119906Visin119864(119866)(119889(119906) + 119889(V)) wasdefined as the first Zagreb index which was one of the mosttraditional indexes in chemical science By means of a largenumber of experimental studies this framework can be usedin each of the physicochemical properties with fixed octanedatabase As an example an evident improvement can beobtained in the octanol-water partition coefficient and itwas pointed out that the absolute value of the correlationcoefficient gets a tight maximum if taking 120582 = minus014 in theabove framework Then by virtue of derivation the octanol-water partition coefficient of octanes can be stated as thefollowing representation

log119875 = minus02058( sum119906Visin119864(119866)

(119889 (119906) + 119889 (V))

minus 014 sum119906Visin119864(119866)

(119889 (119906)2 + 119889 (V)2)) + 75864(4)

where log119875 is the logarithm function of the octanol-waterpartition coefficient This fact implies that the error ofmean absolute percentage is only 006 and the correlationcoefficient can reach to 099896

In real engineering implements themodelsum119906Visin119864(119866)(119889(119906)+119889(V)) + 120582sum119906Visin119864(119866)(119889(119906)2 + 119889(V)2) can be regarded as ageneralized framework For different chemical applicationsadaptive parameter 120582 is a key factor which can be changed tothe optimal value according to the detailed applications Inthe above example 120582 takes minus014 while for other applications120582 can take other values and it is determined by detailedphysical-chemical properties and measured in the chemicalexperiments

In thewhole article themolecular structure ismodeled asa graph119866with vertex set119881(119866) and edge set 119864(119866) where eachvertex represents an atom and each edge denotes a chemicalbond between two atoms A topological index defined on themolecular graph can be considered as a real-valued function119891 119866 rarr R+ which maps each chemical structure to areal score Except the forgotten topological index there areseveral famous indices introduced and applied in chemicalengineering such as Wiener index harmonic index sumconnectivity index and eccentric index (see Gao et al [15ndash19] Yang et al [20] Marana et al [21] and Li et al [22] formore details) The terminologies and notations used but notclearly defined in our paper can be found in Bondy andMutry[23]

Although there have been many advances in degree-based and distance-based indices of molecular graphs thestudies of forgotten topological index for special chemicalmolecular structures are still largely limited For this reasonwe give the exact expressions of forgotten topological indexfor several chemical molecular structures which commonlyappeared in various chemical environments


3m3m minus 1

3m minus 2




119865 (119866) = sum119864119894

sum119906Visin119864119894

(119889 (119906)2 + 119889 (V)2)= sum119864lowast119895


(119889 (119906)2 + 119889 (V)2) (5)


119865 (119866) = sum119864119894119895

1003816100381610038161003816100381611986411989411989510038161003816100381610038161003816 (1198942 + 1198952) (6)








Theorem 1 One has




into two partitions





Theorem 2 One has

119865 (TUC4 [119898 119899]) = 64119898119899 minus 10119898 (8)







Theorem 3 One has


(9)










Theorem 4 One has

119865 (VC5C7 [119901 119902]) = 432119901119902 + 48119901119865 (HC5C7 [119901 119902]) = 216119901119902 + 40119901 (10)







12

3

2

3

m

n



middotmiddotmiddot

middotmiddotmiddot

m minus 1

n minus 1


Theorem 5 One has

119865 (TUZC6 [119898 119899]) = 54119898119899 + 16119898119865 (TUAC6 [119898 119899]) = 54119898119899 + 16119898 (11)













119865 (119871119899) = 54119899 minus 22119865 (119885119899) =

32 119899 = 172119899 minus 58 119899 ge 2

(12)







119865 (1198611119899) = 54119899 minus 4 (14)




119865 (1198612119899) = 54119899 + 18119898 minus 40 (15)





119865 (1198613119899) = 18119898 + 54119899 minus 40 (16)


119865 (1198614119899) = 18119898 + 54119899 minus 40 (17)


4 Conclusion








Competing Interests


Acknowledgments


References



































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


3m3m minus 1

3m minus 2




119865 (119866) = sum119864119894

sum119906Visin119864119894

(119889 (119906)2 + 119889 (V)2)= sum119864lowast119895


(119889 (119906)2 + 119889 (V)2) (5)


119865 (119866) = sum119864119894119895

1003816100381610038161003816100381611986411989411989510038161003816100381610038161003816 (1198942 + 1198952) (6)








Theorem 1 One has




into two partitions





Theorem 2 One has

119865 (TUC4 [119898 119899]) = 64119898119899 minus 10119898 (8)







Theorem 3 One has


(9)










Theorem 4 One has

119865 (VC5C7 [119901 119902]) = 432119901119902 + 48119901119865 (HC5C7 [119901 119902]) = 216119901119902 + 40119901 (10)







12

3

2

3

m

n



middotmiddotmiddot

middotmiddotmiddot

m minus 1

n minus 1


Theorem 5 One has

119865 (TUZC6 [119898 119899]) = 54119898119899 + 16119898119865 (TUAC6 [119898 119899]) = 54119898119899 + 16119898 (11)













119865 (119871119899) = 54119899 minus 22119865 (119885119899) =

32 119899 = 172119899 minus 58 119899 ge 2

(12)







119865 (1198611119899) = 54119899 minus 4 (14)




119865 (1198612119899) = 54119899 + 18119898 minus 40 (15)





119865 (1198613119899) = 18119898 + 54119899 minus 40 (16)


119865 (1198614119899) = 18119898 + 54119899 minus 40 (17)


4 Conclusion








Competing Interests


Acknowledgments


References



































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of




Theorem 3 One has


(9)










Theorem 4 One has

119865 (VC5C7 [119901 119902]) = 432119901119902 + 48119901119865 (HC5C7 [119901 119902]) = 216119901119902 + 40119901 (10)







12

3

2

3

m

n



middotmiddotmiddot

middotmiddotmiddot

m minus 1

n minus 1


Theorem 5 One has

119865 (TUZC6 [119898 119899]) = 54119898119899 + 16119898119865 (TUAC6 [119898 119899]) = 54119898119899 + 16119898 (11)













119865 (119871119899) = 54119899 minus 22119865 (119885119899) =

32 119899 = 172119899 minus 58 119899 ge 2

(12)







119865 (1198611119899) = 54119899 minus 4 (14)




119865 (1198612119899) = 54119899 + 18119898 minus 40 (15)





119865 (1198613119899) = 18119898 + 54119899 minus 40 (16)


119865 (1198614119899) = 18119898 + 54119899 minus 40 (17)


4 Conclusion








Competing Interests


Acknowledgments


References



































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


12

3

2

3

m

n



middotmiddotmiddot

middotmiddotmiddot

m minus 1

n minus 1


Theorem 5 One has

119865 (TUZC6 [119898 119899]) = 54119898119899 + 16119898119865 (TUAC6 [119898 119899]) = 54119898119899 + 16119898 (11)













119865 (119871119899) = 54119899 minus 22119865 (119885119899) =

32 119899 = 172119899 minus 58 119899 ge 2

(12)







119865 (1198611119899) = 54119899 minus 4 (14)




119865 (1198612119899) = 54119899 + 18119898 minus 40 (15)





119865 (1198613119899) = 18119898 + 54119899 minus 40 (16)


119865 (1198614119899) = 18119898 + 54119899 minus 40 (17)


4 Conclusion








Competing Interests


Acknowledgments


References



































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of





119865 (1198613119899) = 18119898 + 54119899 minus 40 (16)


119865 (1198614119899) = 18119898 + 54119899 minus 40 (17)


4 Conclusion








Competing Interests


Acknowledgments


References



































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of

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