New Insights into Aromaticity through Novel Delta Polynomials and Delta Aromatic Indices

: We have developed novel polynomials called delta polynomials, which are, in turn, derived from the characteristic and matching polynomials of graphs associated with polycyclic aromatic compounds. Natural logarithmic aromatic indices are derived from these delta polynomials, which are shown to provide new insights into the aromaticity of polycyclic aromatic compounds, including the highly symmetric C 60 buckminsterfullerene, several other fullerenes, graphene, kekulene series and other cycloarenes, such as polycyclic circumcoronaphenes and coronoids. The newly developed aromatic index yields a value of 6.77 for graphene, 6.516865 for buckminsterfullerene C 60 (I h ), 5.914023 for kekulene (D 6h symmetry), 6.064420 for coronene (D 6h ), 6.137828 for circumcoronene (D 6h ), 6.069668 for dicronylene and so forth. Hence, the novel scaled logarithmic aromatic delta indices developed here appear to provide good quantitative measures of aromaticity, especially when they are used in conjunction with other aromatic indicators.


Introduction
The concept of aromaticity has intrigued both experimental [1][2][3][4][5][6][7][8][9][10][11][12] and theoretical chemists  resulting in a plethora of publications on the topic over the decades.The landscape of aromatic compounds has dramatically metamorphosed with the advent of molecules such as the highly symmetric buckminsterfullerene [1]; various fullerenes [2]; and circumcised coronoidal polycyclic aromatics, such as kekulene [3][4][5], septulene [6], octulene [7], porous nanographenes [9][10][11][12][13][14][15][16] and so forth.Consequently, the old concepts of aromaticity that included primarily planar polycyclic compounds with 4 n + 2 π-electrons has evolved into intriguing concepts such as the 3D-aromaticity, spherical aromaticity, superaromaticity, etc., and thus encompass non-planar compounds and even compounds that do not strictly conform to the 4n + 2 π-electron rule.Circumcised coronoidal polycyclic aromatic compounds that display extended macrocyclic conjugation such as circumkekulene, non-alternant septulene [6], nanographenes [9][10][11][12][13][14][15][16] and the truncated icosahedral C 60 with I h symmetry [1,2], the cynosure of fullerenes, have all contributed to the evolution of the topics of aromaticity and superaromaticity to encompass such a large array of varied compounds in striking contrast to planar polycyclics with 4n + 2 π-electrons .Consequently, the aromaticity concept continues to challenge our understanding owing to the enhanced thermodynamic stability of several of these polycyclic compounds that can only be explained through the generalization of these concepts to encompass the phenomenon of superaromaticity and spherical aromaticity.Yet aromaticity continues to be an elusive concept, defying our established conceptual platforms and pointing to the compelling requirement for the development of novel ideas to encompass such a varied platform of polycyclic compounds that exhibit enhanced thermodynamic or kinetic stabilities.
The advent of kekulene [3][4][5], a circumcised coronene with D 6h symmetry, demonstrates the existence of a structure with a cavity made possible by an assembly of angularly annulated benzene rings which arises from a combination of two interacting [4n + 2]annulenes.The enhanced thermodynamic stability of kekulene is experimentally demonstrated Symmetry 2024, 16, 391 2 of 26 with the synthesis of this molecule and the observed proton NMR chemical shifts and magnetic properties-all of which suggest that the extended ring currents arise from individual benzene rings [3][4][5], as opposed to overly extended delocalizations around the entire structure.Furthermore, sister polyarene molecules with cavities such as septulene and octulene have been synthesized over the years [6,7], although septulene, with a seven-fold symmetry exhibiting the D 7h point group, does not conform to the traditional notion of an alternant polycyclic aromatic compound.Notwithstanding the fact that septulene [6] is not alternant and does not conform to the typical [4n + 2] rule, it exhibits electronic and magnetic properties that are reminiscent of kekulene, provoking us to revisit our conventional notions of aromatic compounds.
Topological, group theoretical and graph theoretical techniques  have been developed and applied to a large number of polycyclic aromatic compounds, organic polymers, fullerenes, circumcised coronoidal structures with cavities, nanographenes and so forth with the objective of shedding light on their structures, spectra, combinatorial chemistry, properties, magnetic and electronic features, aromaticity and toxicity.One such technique that has enjoyed considerable success is the conjugated circuit theory [42][43][44]65], which relies on the combinatorial enumeration of inherent conjugated circuits and Clar's aromatic sextets [19,28,40].The technique has facilitated a reliable platform for understanding the relative stabilities, aromaticity and magnetic and electronic properties of polycyclic aromatics.Furthermore, such combinatorial and graph theoretical methods have provided significant new insights into intriguing phenomena such as superaromaticity, which is a form of macrocyclic aromaticity.These techniques have revealed that the macrocyclic conjugation inherent to these structures is the primary cause of their enhanced thermodynamic stability.Combinatorics and graph theory have been applied to the enumeration of conjugated circuits, isomers of polycyclic aromatics and their derivatives, spectral polynomials, matching polynomials, distance polynomials and a number of polycyclic aromatics and fullerene cages .An intriguing feature of such applications is that some of these symmetry-based techniques involve such novel group theoretical techniques, such as Euler's totient functions, Polya's theory of enumeration, etc., to predict their polysubstituted isomers and spectra [77].Many of these polysubstituted aromatics, macrocyclic arenes, heteropolycyclic arenes and related halocarbons have been studied owing to their significant importance as environmental pollutants, carcinogens, hepatotoxins, industrial chemicals and petroleum products.Furthermore, macrocyclic compounds find applications in the environmental remediation through the selective sequestration of metal ions, and consequently, they find important applications in the sequestration of toxic metal ions, for example, in high-level nuclear wastes.Hosoya [21] has carried out pioneering studies connecting symmetry and matchings of graphs and extensive work has been carried out by Hosoya [18][19][20][21][22][23][24], Aihara [13,23,25,26,[29][30][31][32][33]35,40,[49][50][51][52][53][54][55], 53,[56][57][58] and the author and coworkers [16,[66][67][68][69][70][71][72][73][74][75][76] connecting such polynomials, graph theoretical concepts, resonance energies and so forth to gain insights into aromaticity.
The above survey of experimental and theoretical interest in aromaticity and polycyclic compounds clearly demonstrates significant interest in the topic and the somewhat elusive nature of aromaticity.Despite all these developments, aromaticity continues to baffle researchers in this field due to the varied complexity of compounds that belong to this class.Stimulated by several pioneering conceptual studies that we have cited herein, the present work extends several of these ideas to encompass both characteristic and matching polynomials to evolve into novel aromatic scaled delta and zeta indices together with delta polynomials.We have knitted many of these concepts into a novel fabric of aromaticity in order to apply these concepts to a vast array of polycyclics, including three-dimensional fullerenes and polycyclic structures containing cavities as well as conventional polycyclic aromatics.We demonstrate the utility of delta polynomials and the novel indices that we propose in this study for a variety of such compounds.

Delta Polynomials: Mathematical and Computational Methods
We start with the definition of the adjacency matrix of a graph: 1 i f vertices i and j are connected 0 otherwise An important graph-theoretical invariant, although not unique, is the characteristic polynomial of the graph, denoted by P G .The characteristic or the spectral polynomial of a graph is given by: where C k , is the kth coefficient in the characteristic polynomial, which can be interpreted combinatorically through Sachs' theorem: G i s is Sachs' subgraph of G containing k vertices; c(g) is the disconnected components in g; and r(g) is the number of cycles in the subgraph g.The related matching polynomial or the acyclic polynomial of a graph G can be defined with p(G, k), which is the number of ways to place k disjoint dimers on the graph G: where [n/2] is the greatest integer contained in n/2.For any graph containing an even number of vertices, the coefficients of the odd terms are zero, and consequently, they are not included in the above definition of the matching polynomials.We also note that the constant coefficient in the matching polynomial enumerates the number of Kekulé structures or full coverings with matching for any graph G. Another way to express the matching polynomial that would include zero alternating coefficients is: where a k is zero for odd terms while it is the number of dimers for even terms.Hence, a k is sometimes called the acyclic coefficient, while M G (x) is also referred to as the acyclic polynomial as it contains the acyclic components of Sachs' subgraphs in G.The above definition is more convenient to compare the matching polynomial and the characteristic polynomial of a graph.The spectra of a graph G are simply the eigenvalues of the adjacency matrix or the roots of the characteristic polynomial as defined above.Likewise, the roots of the matching polynomial constitute the matching spectra of G.For several graphs, the present author [74] showed that the matching spectra are the eigenvalues of complex-edge-weighted graph derived from G by assigning complex weights so that the overall adjacency matrix is hermitian.Moreover, Hosoya and the author [78,79] have shown that the matching polynomials of a number of graphs can be obtained as the characteristic polynomials of complex-edge weighted graphs, although these techniques are restricted to a few graphs and the weighting scheme becomes more complicated for larger graphs containing multiple fused cycles.As shown by Aihara [33], an important insight can be obtained into aromaticity through the concept of topological resonance energy, which is obtained as the weighted algebraic sum of the difference between graph spectral eigenvalues and the matching spectral values.Although this is an important measure of aromaticity, it is a difficult quantity to deal with as the matching spectra of graphs containing a large number of vertices with several fused cycles as in polycyclic aromatic compounds are difficult to obtain, although the graph spectra can be more readily obtained by diagonalizing the adjacency matrix by invoking symmetry or by the standard Givens-Householder tri-diagonalization technique.Even for graphs containing a very large number of vertices, it has been shown that the Hadamard transform technique can be employed to extract all eigenvalues of the adjacency matrix [80].
Consequently, the bottleneck of the topological resonance energy computation lies in the computation of matching spectra for highly clustered graphs containing large numbers of vertices.Although Aihara [33] suggested the use of bond resonance energy and circuit resonance energy to circumvent this difficulty, the quantitative measures of aromaticity continue to pose challenges for graphs containing multiple fused cycles with a large number of vertices.
Stimulated by the works of Hosoya [18][19][20][21][22][23][24] as well as Hosoya and the current author [78,79], we propose here novel polynomials which we call delta polynomials and derive natural logarithmic aromatic indices from the coefficients of delta polynomials.The delta polynomial for any graph is defined as follows: where C k and a k are the coefficients from the characteristic polynomial and matching polynomial, respectively.We note that the coefficients in the delta polynomial are always non-negative, and the first few terms of the delta polynomial tend to be zero.Moreover, for trees or acyclic polyenes, all coefficients in the delta polynomials are identically zero because the matching and characteristic polynomials become identical for trees.As seen from Sachs' theorem, the coefficients of the characteristic polynomials contain both cyclic and acyclic components while the coefficients of the matching polynomials contain purely acyclic components.Consequently, delta polynomials contain all important cyclic components of various lengths together with some dimeric components, and thus include several important measures required to provide quantitative measures of aromaticity.However, as these coefficients tend to increase in magnitude sharply for larger graphs, we define two aromatic indices based on the coefficients of delta polynomials.
where the sum is taken over all non-zero coefficients of the delta polynomial and n is the number of vertices in the graph.The second aromatic index is considered a weighted index, as it includes the lengths of various components that are purely non-acyclic.Note that for comparison, Hosoya's Z index [18,21,22] and the related Z c index are obtained from the coefficients of the matching and characteristic polynomials as defined by: As both Z and Z c grow astronomically, and in order to make them comparable to our delta aromatic indices, we introduce two indices using the scaled natural logarithmic functions as follows: where the first zeta-index is obtained using the coefficients of the matching polynomial, while the second zeta-index is obtained from the coefficients of the characteristic polynomials.Consequently, we have four measures that can be computed and compared for different graphs.Among these, we have found that both regular and weighted delta indices are good predictors of aromaticity and the relative order of aromaticity among a variety of compounds that we compare here.
The characteristic polynomials of all structures were computed using the Frame method developed previously [71][72][73][74] while the matching polynomials were computed using a recursive pruning algorithm described in detail in previous studies [71][72][73][74].We note that the philosophy behind the delta index in comparison to the zeta indices is that the zeta index derived from the matching polynomial includes only acyclic contributions while the zeta index obtained from the characteristic polynomial includes both cyclic and acyclic contributions without any differentiation.Therefore, the delta indices offer a compromise as they completely eliminate purely acyclic components.The other advantage of the delta indices is that unlike the topological resonance energy or bond energy or circuit energy computations that require the eigenvalues, the delta indices are easier to compute as they involve only the coefficients of the matching and characteristic polynomials.This is an advantage because for larger systems, the computations of all roots of the matching polynomials with reasonable accuracy could pose problems.It appears that the delta indices offer a reasonable comprise and yet they seem to closely mimic the aromaticity trends.It should be noted that the delta polynomials go to zero for trees or purely acyclic molecules, which is consistent with the fact that such compounds are not aromatic, and hence the delta indices are not defined for such purely acyclic molecules that are not aromatic.The next section describes the results of our computations and comparison of a number of polycyclic compounds with varied complexity, including three-dimensional structures such as fullerenes C 60 and C 70 .

Results and Discussion
We considered a number of structures with varied complexities for the study of delta polynomials and the zeta and delta aromatic indices of these structures.Figure 1 shows a compilation of such structures that were considered in this study.As seen from Figure 1, we included planar polycyclic compounds and three-dimensional spherical structures such as C 60 , C 70 and C 72 , as well as polycyclic structures with holes, such as kekulene, septulene and a zigzag macrocycle containing 21 rings.Consequently, these structures offer quite a varied platform for the comparison of relative aromaticity on the basis of the various computed indices.
The characteristic polynomials of all structures were computed using the Fra method developed previously [71][72][73][74] while the matching polynomials were compu using a recursive pruning algorithm described in detail in previous studies [71][72][73][74].note that the philosophy behind the delta index in comparison to the zeta indices is t the zeta index derived from the matching polynomial includes only acyclic contributio while the zeta index obtained from the characteristic polynomial includes both cyclic a acyclic contributions without any differentiation.Therefore, the delta indices offer a co promise as they completely eliminate purely acyclic components.The other advantage the delta indices is that unlike the topological resonance energy or bond energy or circ energy computations that require the eigenvalues, the delta indices are easier to comp as they involve only the coefficients of the matching and characteristic polynomials.T is an advantage because for larger systems, the computations of all roots of the match polynomials with reasonable accuracy could pose problems.It appears that the delta dices offer a reasonable comprise and yet they seem to closely mimic the aromatic trends.It should be noted that the delta polynomials go to zero for trees or purely acy molecules, which is consistent with the fact that such compounds are not aromatic, a hence the delta indices are not defined for such purely acyclic molecules that are not a matic.The next section describes the results of our computations and comparison o number of polycyclic compounds with varied complexity, including three-dimensio structures such as fullerenes C60 and C70.

Results and Discussion
We considered a number of structures with varied complexities for the study of de polynomials and the zeta and delta aromatic indices of these structures.Figure 1 show compilation of such structures that were considered in this study.As seen from Figure we included planar polycyclic compounds and three-dimensional spherical structu such as C60, C70 and C72, as well as polycyclic structures with holes, such as kekulene, s tulene and a zigzag macrocycle containing 21 rings.Consequently, these structures off quite a varied platform for the comparison of relative aromaticity on the basis of the va ous computed indices.Table 1 illustrates the computation of various aromatic indices; we have included th characteristic and matching polynomials of isomers of very simple structures with thre benzene rings, that is, phenanthrene and anthracene.As can be seen from Table 1, as bot are purely alternant benzenoids, as characterized by their bipartite graphs, the coefficient of the odd terms are zeroes.The constant term of the matching polynomial is simply th Table 1 illustrates the computation of various aromatic indices; we have included the characteristic and matching polynomials of isomers of very simple structures with three benzene rings, that is, phenanthrene and anthracene.As can be seen from Table 1, as both are purely alternant benzenoids, as characterized by their bipartite graphs, the coefficients of the odd terms are zeroes.The constant term of the matching polynomial is simply the number of resonance structures, which is five for phenanthrene and four for anthracene, a well-known result, indicating that phenanthrene is more aromatic than anthracene.Herndon's [81] resonance energy is simply formulated as a scaled log of the number of resonance structures multiplied with a constant.However, we note that the constant term in the matching polynomial alone does not correlate with aromaticity or the stability of a structure.One needs to consider the contributions from various circuits, which are included in the other coefficients.The coefficients of other terms in the two polynomials yield the combinatorial numbers for other Sachs' subgraphs.The delta polynomials thus contain only non-acyclic terms enumerated among the Sachs' subgraphs, although some of the terms contain both disjoint circuits and dimers.The last but one row in Table 1 shows the sum of the coefficients of the characteristic polynomial, the sum of the coefficients of the matching polynomial, the well-known Hosoya's [18,22] Z index and, finally, the corresponding sums of delta polynomial coefficients.We also introduced a new weighted index concept that we designate as ∑ kδ k , which weighs over different components of the Sachs' non-acyclic graphs.The philosophy behind this is that not all Sachs' circuits contained in different coefficients make the same contribution, and hence one needs to introduce weights as given by the total number of vertices in these disjointed circuits or simply k.For example, k = 8 would designate a circuit of length 8, a circuit of length 6 + a dimer, and so forth.Thus, by weighting the coefficients with k, we have taken this important factor into account, that is, not all coefficients have the same circuit lengths, and thus, the weighting method provides a means for contrasting their contributions.The last row shows the scaled natural logarithmic indices derived from these sums.First, the natural logarithmic functions reduces the astronomically large combinatorial numbers for the sums of these coefficients for large polycyclics.This, combined with a scaling method, in which we divide the natural logarithm by the number of vertices, eliminates the size dependency.Thus, the scaled logarithmic index provides a uniform basis to compare and contrast the aromaticity of a large number of polycyclic compounds with varied sizes and complexities.Therefore, as can be seen from Table 1, phenanthrene has scaled zeta and delta indices of ζ C : 0.5218471, ζ M : 0.5083718, ∆:0.3960841 and ∆ w : 0.5642518, while the corresponding indices for anthracene are ζ C : 0.5194570, ζ M : 0.5069087, ∆: 0.3890527 and ∆ W : 0.5559446.It was noted that the indices reveal a contrast between phenanthrene and anthracene and predict a correct trend of aromaticity.However, in general cases, as the Hosoya index is derived from purely acyclic or dimer terms, while aromaticity involves circuits, delta indices, especially in the weighted forms, offer a good measure of aromaticity.This is especially true when a comparison needs to be made for compounds of varied sizes and complexities.All techniques lead to the same conclusion that phenanthrene is more aromatic than anthracene, as expected.We also obtained the delta polynomials of a number of zigzag and linear polyacenes of larger sizes.The general trend is that the weighted delta index is larger for the zigzag structures compared to linear polyacenes, which is consistent with the trend that the zigzag polyacenes are more aromatic than linear polyacenes.This arises from a Fibonacci increase in the resonance count for each kink in the structure of a zigzag polyacene.This in turn translates into a larger weighted delta index for a zigzag polycyclic as compared with a linear polyacene.The two simple cases are considered for illustrative purposes only, as we demonstrate the power of the techniques with more complex polycyclics starting with coronene and circumcoronene.The computed results for these two structures are shown in Tables 2 and 3, respectively.As seen from these tables, the first several terms of the delta polynomials are zeroes, as these terms contain only purely acyclic contributions.For coronene, the first non-zero term in the delta polynomial corresponds to the seventh term, which contains the contributions of from a hexagon in the structure, and since there are no four-membered rings in the structure, only hexagons make contributions to this term.Starting with this term, all other subsequent terms contain various other types of circuits in the structure together with contributions from some disjoint dimers.Consequently, the unweighted delta indices computed from the coefficients of the delta polynomial are 0.4897992 and 0.5482407, respectively, clearly suggesting that circumscribing coronene results in considerably enhanced aromaticity.The constant coefficients of the two matching polynomials are given by 20 and 980, respectively, which are also the well-known Kekulé counts of the two structures.As these are purely alternant benzenoid polycyclic aromatic compounds, the square of the constant coefficients in the matching polynomials yields the constant coefficients in the characteristic polynomial.We shall discuss in a subsequent paragraph the weighted delta index together with the other indices of all polycyclics considered in this study.Table 4 shows the various polynomials obtained for kekulene together with the corresponding unweighted zeta and delta indices.Kekulene can be generated by the circumcision of the central hexagon of the coronene structure.Consequently, when one compares the unweighted delta indices of the two structures, one can infer that circumcision results in a lower delta index for kekulene compared to coronene.That is, circumcision disrupts the various circuits that were present in coronene, resulting in a lower π-electronic ring current or lower aromaticity in kekulene compared to coronene.This feature is mirrored by the delta indices of the two structures.

Circumbezo[ghi] perylene Hexabenzocoronene
Tables 5 and 6 consider three polycyclic isomers of C 22 H 12 and the corresponding circumscribed structures of the three isomers, respectively.The three isomers have been enumerated in the handbook by Dias [57] on polycyclic aromatic compounds.The three isomers, namely, triangulene, anthanthrene and benzo[ghi]perylene, represent interesting cases for our study.Among these, triangulene exhibits a triplet electronic ground state and it is thus a diradical.We note that this is directly inferred by the zero coefficient of the constant term of the characteristic polynomial of triangulene, consistent with a doubly degenerate set of HOMOs, resulting in a triplet ground state.As seen from Table 5, the delta indices of the three structures indicate that bezo[ghi]pyrelene is the most aromatic of the three structures with triangulene being the least aromatic.Although the combinatorial numbers in Table 6 become more complex due to a greater number of various circuits in the corresponding circumscribed structures, the final delta indices are much more amenable to critical comparison and thus shed light on the potential aromaticity of these compounds.Again, comparing the delta indices of the primitive and circumscribed counterparts in Tables 5 and 6 reveals that circumscribing results in greater aromaticity compared to the uncircumscribed structure.The gaps relative to aromaticity among the isomers are narrowed somewhat when one compares the circumscribed structures to their uncircumscribed counterparts.In particular, triangulenes obtain a greater aromaticity upon circumscribing.However, we note that better measures are obtained using the weighted delta indices which we compare in a subsequent Table and paragraph.Furthermore, as seen from Table 6, the constant coefficient of the circumscribed triangulene continues to be zero for the characteristic polynomial suggesting that circumtriangulene continues to exhibit a triplet diradical ground state although its aromaticity is enhanced relative to the primitive triangulene structure.This trend is repeated by a number of structures that we have tested and are not shown here.The general trend is that circumscribing a given structure results in enhanced aromaticity while circumcision results in lower aromaticity.Next, we consider three-dimensional and other structures that appear to exhibit aromatic characters or unusual stabilities.The celebrated buckminsterfullerene with the icosahedral I h group has been the cynosure of fullerenes.Table 7 displays all three polynomials of C 60 together with the sums of the columns and the scaled natural log indices.There are several differences that should be noted in the polynomials for the C 60 structure compared to the other polycyclics that we have considered thus far.None of the coefficients of the odd terms except the first two odd terms is zero for the characteristic and delta polynomials of C 60 .For example, the sixth or fifth coefficient, not counting the first term, is twenty-four for the delta polynomial and is consistent with the twelve pentagons present in the structure.Likewise, all other odd terms are non-zeroes and contribute toward the delta polynomial.This is a striking contrast compared to alternant benzenoid hydrocarbons.Furthermore, the square of the constant term in the matching polynomial does not yield the constant term of the characteristic polynomial.These features, together with several non-zero odd terms in the delta polynomials, provide C 60 with some unique features.Although the sum of all the coefficients of the delta polynomial is 2508935631291784, a large number, the scaled log of the corresponding result is 0.5909773, suggesting that its unweighted delta aromaticity is much higher than the circumcoronene value of 0.5482407 and even circumcircumcoronene, circumkekulene and so forth.The weighted delta aromatic indices yield further insights, as we will now discuss.These results of C 60 can also be compared with the corresponding indices of other fullerenes, such as C 70 (D 5h ) and C 72 (C 2v ), as shown below.These computations can be extended to large macrocycles, such as the one shown in Figure 1 containing 21 hexagons arranged in a zigzag fashion with a large internal cavity.The various polynomials for such a macrocycle are shown in Table 8.As this is an alternant benzenoid hydrocarbon, we show only the coefficients of the even terms as all odd terms have zero coefficients for all three polynomials.This macrocycle with a zigzag structure has a delta index of 0.5352335, making it comparable to circumcoronene, which has a delta index of 0.5482407.This implies that the zigzag macrocycle with 21 rings less aromatic compared to circumcoronene but more aromatic compared to kekulene with a delta index of 0.5203879.We note that the weighted delta index appears to provide yet another reliable way to compare different structures, although any of these indices should be used in conjunction with other parameters, such as the HOMO-LUMO gap or electronic or magnetic parameters derived from quantum chemical computations.
Table 9 shows a cumulative across-board comparison of all four indices of all structures that are considered in the present study.As both delta and zeta indices are derived from scaled natural log values, we uniformly multiplied the indices by a factor of 10 in Table 9 for comparison.It can be seen from Table 9 that the weighted delta index appears to provide one of the best measures of aromaticity and stability.For example, the weighted delta index of buckminsterfullerene stands out as 6.516865, a maximum among all structures considered here with the exception of graphene.In fact, while the weighted delta index of buckminsterfullerene is higher than both C 70 (D 5h ) and C 72 (C 2v ), this trend is not reproduced by any of the other indices shown in Table 9.We note that both C 72 (C 2v ) and C 72 (D 6d ) structures have been found to be stable isomers [82,83].Moreover, we note that as the weighted delta indices do not vary in large magnitude, and small changes should be considered important as the weighted indices are subtle in their variations.As seen from Table 9, circumcoronene is more aromatic than coronene as well as hexbenzcoronene.Polycyclic structures with cavities, such as coronaphene, circumcoronaphene, kekulene, etc., are less aromatic compared to their parent structures prior to circumcision.On the other hand, kekulene and septulene have a remarkably similar aromaticity, as inferred from their weighted delta indices of 5.914023 and 5.884901, respectively.Of the three C 22 H 12 isomers, triangulene exhibits the least aromaticity while benzo[ghi]perylene exhibits the greatest aromaticity.We also note a few variations in trends, for example, ovalene is predicted to be much less aromatic compared to circumovalene on the basis of zeta and unweighted delta index but the weighted delta index exhibits the same trend but with a smaller contrast.While circumpyrene is predicted to be less aromatic compared to circumovalene on the basis of the zeta and unweighted delta indices, the weighted delta index predicts the opposite trend with a smaller contrast.
Kekulene and septulene are virtually identical relative to the zeta indices, as can be seen from Table 9.The identical values of the scaled Hosoya index require further inquiry.Moreover, the delta index suggests an opposite trend, in that it predicts septulene to be slightly more aromatic compared to kekulene, although the weighted delta index predicts kekulene to be slightly more aromatic than septulene.The sum of the coefficients of the three polynomials and the weighted sum for kekulene are given as 170396692000, 99914817684, 70481874316 and 2130357387264, respectively.The corresponding values for septulene are 12686887009024, 6806150529706, 5880736479318 and 205289991176192, respectively.Consequently, although these numbers are quite different for kekulene and septulene, when they are scaled by the number of vertices after taking log of these values, accidental degeneracy is reached for kekulene relative to Hosoya's Z index while the Z c index is almost the same.Thus, these two indices fail to differentiate septulene and kekulene while the weighted delta index appears to provide the correct trend.
We were able to obtain an estimate of the aromaticity delta index for graphene using an extrapolation scheme by using the results of coronene, circumcoronene and circumcircumcoronene.A previous study on the degree-based topological indices of series of polycyclic aromatics [69] has revealed that if one extrapolates the results of known circumcoronene series with the order of circumscribing as n~6.4,one obtains the results converging to graphene.By using the same extrapolation scheme with the results obtained for coronene, circumcoronene and circumcircumcoronene, we obtain the weighted and unweighted delta indices for graphene converging to 6.77.Consequently, one can compare this value to C 60 value of 6.5116865 and conclude that the correct trend is predicted by the newly formulated delta aromatic indices and delta polynomials.Indeed, the highly symmetric buckminsterfullerene is confirmed to be the most stable species among small molecules, fullerenes, and clusters, which corroborates with experimental observations.Furthermore, we note that other topological indices have been applied to different forms of carbon and other complex networks such as diamond and other metal organic frameworks [84,85].

Conclusions
In this study, we proposed hybrid polynomials called delta polynomials and created two scaled logarithmic indices, which we called delta aromatic indices.These indices combined with the zeta indices, which are also scaled versions, were evaluated for a number of polycyclic structures, including fullerenes, kekulenes, septulene, circumcoronene, circumcoronaphene, dicronylene, macrocycles and different isomers of polycyclic compounds.It was shown that the delta indices, especially the weighted delta indices, appear to conform closely with the aromaticity trends of the investigated compounds.We suggest that these newly proposed delta indices can be used in conjunction with other topological, electronic, magnetic and quantum chemical parameters to gain considerable insights into the longstanding phenomenon of aromaticity, superaromaticity and spherical aromaticity.

Figure 1 .
Figure 1.Structures of polycyclic compounds considered in this study.Not all displayed structures are meant to show any particular resonance/Clar's structure; the structures simply show the relationships and connectivities of various hexagonal rings.

Table 1 .
Characteristic, matching and delta polynomials of phenanthrene and the derived Z, zeta and delta indices.

Table 2 .
Characteristic, matching and delta polynomials of coronene.
a x k designates the coefficient in the respective polynomial (characteristic or matching or delta polynomial) of the corresponding column.

Table 3 .
Characteristic, matching and delta polynomials of circumcoronene.

Table 4 .
Characteristic, matching and delta polynomials of kekulene.

Table 5 .
Characteristic, matching and delta polynomials of three polycyclic isomers of C 22 H 12.

Table 7 .
Characteristic, matching and delta polynomials of C 60 buckminsterfullerene.

Table 8 .
Characteristic, matching and delta polynomials of the zigzag macrocycle-21 together with their indices.