Scientific paper
LEL-a Newly Designed Molecular Descriptor
Dragan Stevanovi},a'b Aleksandar Ili},a Cristina Onisorc and Mircea V. Diudeac'*
a Faculty of Sciences and Mathematics, University of Ni{, Vi{egradska 33, 18000 Ni{, Serbia
b University of Primorska - FAMNIT, Glagoljaška 8, 6000 Koper, Slovenia
c Faculty of Chemistry and Chemical Engineering, "Babes-Bolyai" University, Arany Janos Str. 11,
400084, Cluj, Romania
* Corresponding author: E-mail: diudea@gmail.com
Received: 28-11-2008
Abstract
A correlating study of topological indices TIs provided by TOPOCLUJ software package and LEL, a newly proposed index built up on the eigenvalues of Laplacian matrix, on thirteen properties of octanes, revealed good correlating ability of a dozen selected TIs, all related to the Wiener index, and of LEL as well. LEL describes well the properties which are well accounted by the majority of the selected molecular descriptors: octane number MON, entropy S, volume MV, or refraction MR, particularly the acentric factor AF parameter, but also more difficult properties like boiling point, melting point and logP. LEL is the best correlated with the WK (Wiener-type number, taken the reciprocal of entries in the combinatorial Dp matrix, of higher rank, calculated by TOPOCLUJ software) indices. In a second set of polycyclic aromatic hydrocarbons, LEL was proved to be as good as the Randi} x index (a connectivity index) and better than the Wiener index (a distance based index). In addition, it is well defined mathematically and shows interesting relations in particular classes of graphs, these recommending LEL as a new, powerful topological index. The actual study proved the considered TIs are basic topological descriptors in prediction of various molecular properties, with good perspective in QSPR/QSAR studies.
Keywords; Topological indices, QSPR, QSAR, octanes, polycyclic aromatic hydrocarbons
1. Introduction
Elucidation of the relationship between molecular structures and their properties is a challenge to chemists for more than a century. A chemical structure can be quantified in various ways, one of the most popular, in the last decade being that which makes use of Graph Theory, particularly topological indices TIs, which are numerical descriptors encoding topological attributes of a molecular graph. They were used both in graph discriminating analysis as well as in quantitative structure-property relationship QSPR studies for modeling a variety of physico-chemical properties or in quantitative structure-activity relationship QSAR studies to predict biological activities. Nowadays, their number becomes uncountable, as a consequence of the explosive development of computational technology. Randi}1 has outlined some desirable attributes for the topological indices in the view of
preventing their hazardous proliferation, among which: direct structural interpretation, good correlation with at least one property, good discrimination of isomers, simplicity, locally defined, and generalized to higher analogues, are the most important. There are several commercial software packages which calculate more than one thousand TIs: POLLY,2 MOLCONN,3 CODESSA,45 DRAGON 6 and TOPOCLUJ.7
Alkanes represent an interesting class of compounds as a starting point for the application of molecular modeling procedures. Many properties of the alkanes vary function of molecular mass or branching, and alkanes can be described by using a single type of (carbon) atom. There are properties well accounted by a single molecular descriptor, e.g., octane number MON, entropy S, volume MV, refraction MR, etc. Other properties, such as, boiling point BP, heat of vaporization HV, total surface area TSA, partition coefficient LogP, density DENS, critical tempe-
rature CT, critical pressure CP and heat of formation DHF are notable exceptions, being not well modeled by any of the parameter sets.
Among the well modeled parameters, the acentric factor AF appears to be correlated more than 90% by eleven of twelve descriptors herein discussed. Definition of the best scored TIs, e.g., LEL, 2WD, 2WH, 1WK, 2WUCJD and PDS3[Sh[SZ]] will be given in the following section.
The purpose of the present report is to evaluate the relative performances of a pool of descriptors in relating the hydrocarbon molecular structures to a set of physical properties. It is of particular usefulness to predict values of some molecular properties, namely those which are difficult to measure or show health risk or for unavailable substances. In this respect, the newly designed index LEL and those provided by the TOPOCLUJ software are of basic importance. The sets of studied molecules were selected among the representative and sufficiently complex structures (octane isomers and polycyclic aromatic hydrocarbons (PAH)) and the correlations used were in monovariate regression, in view of a more direct interpretation of the results.
2. Description of Indices
In any process of molecular modeling, either quantum or correlating one, the need for a representation of molecular structure is critical and its role is significant to find appropriate predictive models. An information rich representation which is rapidly computed and readily manipulated is essential. This is the case of the topological indices, which are among the most used molecular descriptors. TIs are single number descriptors associated with a molecular graph representing a molecule, which does not depend on the numbering and pictorial representation of that graph. In this section, definitions for LEL and the best scored TIs, among the indices provided by the TOPOCLUJ software, are presented.
2. 1. LEL - an Index Built on the Laplacian Matrix
Let G = G(V,E) be a finite, undirected graph with n vertices V = {1,2,...,n} and m = |E| edges. The degree of a vertex u in V is denoted by du. Let G have the adjacency matrix A with eigenvalues X1 > X2 > ... > and Laplacian matrix L = D-A, where D is the diagonal matrix of vertex degrees, with eigenvalues > > ••• > ^n. The Laplacian-like energy, shortly LEL, of G is defined8 as:
(1)
It has been shown to have a nice mathematical behavior. It is closely related to the coefficients ck of the cha-
racteristic polynomial of the Laplacian matrix L:
1=0
(2)
In particular,9 for two graphs G and H of order n, if ck(G) < ck(H) for k = 1,...n-1, then LEL(G) < LEL(H). Furthermore, if a strict inequality ck,(G) < ck,(H) holds for at least one value k', then LEL(G) < LEL(H). Using this relation, it has been proved9 that, among trees of order n, the star Sn has the minimum, while the path Pn has the maximum LEL. Similarly, among unicyclic graphs of order n, the star with an edge between two of its leaves has the minimum LEL, and the cycle Cn has the maximum LEL.10
2. 2. Walk Indices or Wiener-type Indices of Higher Rank
A walk w(1,n) in a graph G = G(V,E) (with V = V(G) being the set of vertices and E = E(G) the set of edges) is an alternating sequence of vertices and edges, w(1,n) = (V1, e1, V2, e2, ..., Vn-1, em, Vn), v, e V(G ), e, e E(G ), m > n - 1,so that any two subsequent edges are adjacent: (v, ^, v,) e E(G). Revisiting vertices and edges is allowed.
If V(w(1,n)) = {v1, v2, ..., vn-1, vn} is the set of vertices of the walk w(1,n) and E(w(1,n)) = {e1, e2, ..., em-1, em} the set of its edges, then l(w(1,n)) = |E(w1n)| represents the length of walk w(1,n), which equals the number of traversed edges. If no other condition is imposed, the walk is called a random walk. If the walk starts and ends in the same vertex v^ = v1 it is a closed walk, else it is open.1117
Walks of length e, starting from the vertex i e V (G) can be counted by summing the entries in the row of the e^h power of the adjacency matrix A:
JeV(G)
(3)
is called the walk degree (of rank e) of vertex i (or atomic walk count18,19). Local and global invariants based on walks in graph were considered for correlating with physico-chemical properties.19
Weighted walk degrees can be easily calculated by means of the algorithm proposed by Diudea et al.20 It evaluates a local (topological) property by iterative summation of vertex contributions over all vertices in row i:

r^wM],,=rwM],,=[M],,
(4)
(5)
(6)
where M is a square matrix and eWM is the diagonal matrix of walk degrees (weighted by the property encoded by
the matrix M). The diagonal entries [®WM]ii represent the row sum of entries in the matrix M raised at power e [M^],, or the (weighted by M) walk degrees
(7)
The half sum of all local invariants eWM,^ in G defines a global invariant called the walk number eW^ :

(8)
When M = A, then eWA represents the molecular when M = D (distance matrix), then eWD is
walk count:18
just the Wiener index,21 of rank e. The extension of this idea resulted in Wiener-type indices of higher rank;19 the info matrix M can be any square topological matrix. Among such matrices, the following ones are discussed in the present paper: D, H (of reciprocal topological distance entries, also called Harary matrix),16,17,22 K (of reciprocal Dp matrix entries)16,17,23 and UCJD (the unsymmetrical Cluj matrix on distances).24-26 For definition of these matrices and derived indices the reader is invited to consult the refs.16,17,27
2. 3. Indices Designed on Layer/shell Matrices
Define a layer of vertices located at distance k to the
19,28,29
vertex i as:
G(0,={v vgF(G); d:=k]
(9)
The partition of G with respect to i will be:
with ecc^ being the eccentricity of i (i.e., the largest distance from i to the other vertices of G). The entries in the layer matrix (of vertex property) LM, is defined as:
(11)
with the most used operation being the summation. The zero columns are just the column of vertex properties [LM]i0 = p.. Any atomic/vertex property can be considered as p,. More over, any square matrix M can be taken as info matrix, in supplying local/vertex properties as row sum RS, column sum CS or diagonal entries given by the Walk matrix.16,17
Layer matrix is a collection of the above defined entries:
LM = {[LM],;^; i G F(G); k g [0,1,..,^/(G)] } (12)
with d(G) being the diameter of the graph (i.e., the largest distance in G).
Define the entries in the shell matrix (of pair vertex property) SM as: 29

(13)
with the most used operation being the summation.
Shell matrix is a collection of the above defined entries:
Table 1. Topological indices for octanes
Molecule LEL 1WD 2WD 1WW 1WH 2WH 1WK 2WK 1/1WK 1/2WK 2WUCJD PDS3[Sh[Sz]]
1 2
3
4
5
6
7
8 9
10 11 12
13
14
15
16
17
18
9.153 9.120 9.115 9.114 9.108 9.065 9.079 9.082 9.088 9.056 9.074 9.073 9.049 9.023 9.031 9.020 9.044 8.971
84 79 76 75 72 71 70
74
67
68 67
64 63 66 62
65 58
1848 1628 1512 1476 1360 1316 1280 1312 1420 1176 1208 1172 1072 1032 1128 1000 1096 868
84 79 76 75 72 71
70
71 74
67
68 67
64 63 66 62
65 58
13.743 14.100 14.267 14.317 14.483 14.767 14.733 14.650 14.467 15.033 14.867 14.917 15.250 15.417 15.167 15.500 15.167 16.000
48.279 51.050 52.495 52.947 54.377 56.500 56.317 55.560 53.939 58.878 57.482 57.924 60.792 62.042 59.771 62.799 59.889 67.000
10.564 10.862 10.981 11.014 11.133 11.433 11.367 11.300 11.167 11.633 11.467 11.500 11.800 11.967 11.767 12.033 11.733 12.500
29.040 31.153 32.125 32.411 33.373 35.426 35.024 34.454 33.343 37.107 35.847 36.124 38.493 39.621 37.911 40.191 37.791 43.750
0.095 0.092 0.091 0.091 0.090 0.087 0.088 0.088 0.090 0.086 0.087 0.087 0.085 0.084 0.085 0.083 0.085 0.080
0.034 0.032 0.031 0.031 0.030 0.028 0.029 0.029 0.030 0.027 0.028 0.028 0.026 0.025 0.026 0.025 0.026 0.023
1596 1396 1284 1248 1136 1112 1072 1102 1206 978 1004 968 880 850 940 820 906 706
120.000 78.600 94.320 94.320 110.040 78.600 110.040 94.320 78.600 110.040 125.760 141.480 125.760 78.600 141.480 125.760 141.480 125.760
The zero column, [SM]i0 = 1, in case of zero diagonal square info matrix but any other pair vertex property (written as diagonal entries) can be considered.
The index PDS3 [Sh[SZ]] in Table 1, is calculated as columns sum up to distance three on the shell matrix of Szeged matrix, taken as info matrix. For Szeged matrices and indices the reader is invited to consult refs.161727
The TOPOCLUJ software package7 is designed to calculate topological descriptors from topological matrices and/or polynomials. Several weighting schemes including group electronegativity, group mass and partial charges are enabled. Topological indices derived from the matrices: adjacency, connectivity, distance, detour, distance-
path, detour-path, Cluj, their reciprocal matrices, walk-matrices, walk-operated matrices, layer- and shell-matrices were successfully used in correlating studies and graph discriminating analysis during the last decade.17,30 The values of the best scored TIs for octane isomers are listed in Table 1.
Data for the octanes properties and correlation coefficients, in monovariate linear regression, with the best scored topological indices, among those provided by TOPOCLUJ software, are shown in Table 2. The inter-correlation of indices is presented in Table3 while in Table 4 that of the properties of the octane isomers.
Table 2. Data for octanes properties: boiling point BP, motor octane number MON, heat of vaporizat-ion HV, molar volume MV, entropy S, total surface area TSA, acentric factor AF, molar refraction MR, partition coefficient n-octanol/water Log P , density D, critical temperature CT, critical pressure CP and heat of formation DHF and their correlation coefficients, in monovariate linear regression, with the selected topological indices
Mol.	BP	MON	HV	MV	S	TSA	AF	MR	LogP	DENS	CT	CP	DHF
1	9.15	-	34.41	162.61	111.67	415.30	0.40	39.19	3.67	0.70	296.20	24.64	4.14
2	9.12	23.10	33.81	163.65	109.84	407.85	0.38	39.23	3.61	0.70	288.00	24.80	3.06
3	9.11	35.00	33.89	161.85	111.26	397.34	0.37	39.10	3.61	0.71	292.00	25.60	3.29
4	9.11	39.00	33.89	162.12	109.32	396.04	0.37	39.12	3.61	0.70	290.00	25.60	4.00
5	9.11	52.40	33.61	160.08	109.43	379.04	0.36	38.94	3.61	0.71	292.00	25.74	3.59
6	9.06	77.40	32.26	164.29	103.42	405.11	0.34	39.25	3.65	0.70	279.00	25.60	2.56
7	9.08	78.90	33.20	160.41	108.02	384.93	0.35	38.98	3.54	0.71	293.00	26.60	4.23
8	9.08	69.90	32.59	163.09	106.98	388.11	0.34	39.13	3.54	0.70	282.00	25.80	2.80
9	9.09	55.70	32.64	164.72	105.72	395.08	0.36	39.26	3.54	0.69	279.00	25.00	2.50
10	9.06	83.40	32.47	160.89	104.74	389.79	0.32	39.01	3.65	0.71	290.84	27.20	3.17
11	9.07	81.70	33.28	158.65	106.59	376.91	0.34	38.85	3.54	0.72	298.00	27.40	4.97
12	9.07	88.10	-	158.81	106.06	368.10	0.33	38.84	3.54	0.72	295.00	27.40	5.08
13	9.05	88.70	32.79	157.04	101.48	366.99	0.31	38.72	3.65	0.73	305.00	28.90	4.76
14	9.02	99.90	32.01	159.52	101.31	371.75	0.30	38.93	3.58	0.72	294.00	28.20	4.09
15	9.03	100.00	31.01	165.10	104.09	392.19	0.31	39.26	3.58	0.69	271.15	25.50	3.13
16	9.02	99.40	32.34	157.30	102.06	377.40	0.29	38.76	3.58	0.73	303.00	29.00	4.52
17	9.04	95.90	32.73	158.85	102.39	368.93	0.32	38.87	3.48	0.72	295.00	27.60	4.32
18	8.97	-	31.42	138.60	93.06	390.47	0.26	-	3.62	0.82	270.80	24.50	4.88
													
LEL	0.673	-0.941	0.887	0.647	0.950	0.509	0.991	0.462	0.175	-0.646	0.243	-0.399	-0.319
1WD	0.559	-0.958	0.715	0.627	0.878	0.721	0.966	0.672	0.701	-0.612	0.054	-0.577	-0.459
2WD	0.561	-0.966	0.803	0.595	0.864	0.732	0.956	0.648	0.719	-0.580	0.053	-0.583	-0.424
1WW	0.540	-0.957	0.795	0.628	0.878	0.721	0.966	0.669	0.258	-0.613	0.041	-0.579	-0.453
1WH	-0.573	0.953	-0.826	-0.676	-0.929	-0.622	-0.992	-0.616	-0.679	0.666	-0.111	0.505	0.428
2WH	-0.557	0.948	-0.816	-0.687	-0.929	-0.625	-0.991	-0.633	-0.173	0.676	-0.098	0.510	0.443
1WK	-0.589	0.949	-0.837	-0.682	-0.940	-0.587	-0.995	-0.588	-0.671	0.675	-0.139	0.477	0.410
2WK	-0.569	0.945	-0.825	-0.694	-0.939	-0.594	-0.995	-0.610	-0.153	0.685	-0.121	0.486	0.429
1/1WK	0.599	-0.957	0.843	0.653	0.930	0.611	0.992	0.585	0.187	-0.644	0.130	-0.501	-0.398
1/2WK	0.595	-0.963	0.836	0.623	0.910	0.648	0.983	0.599	0.211	-0.612	0.104	-0.537	-0.395
2WUCJD	0.551	-0.963	0.794	0.592	0.855	0.744	0.950	0.658	0.716	-0.576	0.040	-0.592	-0.431
PDS3													
[Sh[SZ]]	0.432	0.561	-0.010	-0.984	-0.500	-0.896	-0.600	-0.991	-0.122	0.985	0.894	0.948	0.866
Legend of indices : LEL = Laplacian-like energy
1WD and 2WD = Wiener numbers of higher rank (1 and 2) 1WW = Hyper-Wiener number
1WH and 2WH = Harary numbers of higher rank (1 and 2)
1WK and 2WK = Wiener-type number, taken the reciprocal of entries in the Dp matrix, of higher rank (1 and 2).
1/1WK and 1/2WK = global reciprocal of 1WK and 2WK
2WUCJD = Wiener-type number (of rank 2), calculated on Cluj matrix on distance
PDS3 [Sh[SZ]] = Columns sum up to distance 3 on the Shell matrix calculated on SZ matrix
Table 3. Intercorrelation matrix for the best scored indices in octanes
Variable	LEL 1WD 2WD	1WW	1WH	2WH	1WK	2WK	1/1WK	1/2WK	2WUCJD	PDS3
										[Sh[SZ]]
LEL	1.00 0.96 0.95	0.96	-0.99	-0.98	-0.99	-0.99	0.99	0.98	0.94	-0.34
1WD	1.00 1.00	1.00	-0.99	-0.99	-0.98	-0.98	0.99	0.99	1.00	-0.43
2WD	1.00	1.00	-0.98	-0.98	-0.97	-0.97	0.98	0.99	1.00	-0.42
1WW		1.00	-0.99	-0.99	-0.98	-0.98	0.99	0.99	1.00	-0.43
1WH			1.00	1.00	1.00	1.00	-1.00	-1.00	-0.98	0.40
2WH				1.00	1.00	1.00	-1.00	-0.99	-0.98	0.41
1WK					1.00	1.00	-1.00	-0.99	-0.97	0.38
2WK						1.00	-1.00	-0.99	-0.97	0.39
1/1WK							1.00	1.00	0.98	-0.38
1/2WK								1.00	0.99	-0.39
2WUCJD									1.00	-0.43
PDS3[Sh[S]]										1.00
Table 4. Intercorrelation matrix for the selected molecular properties of octanes										
Variable	BP MON HV	MV	S	TSA	AF	MR	LogP	DENS	CT CP	DHF
BP MON HV MV
S
TSA
AF
MR
LogP
DENS
CT
CP
DHF
1.00
-0.32 1.00
0.11 0.08 1.00
0.12 -0.38 -0.02 1.00
0.62 -0.62 0.09 0.73 1.00
0.06 -0.39 -0.32 0.29 0.41 1.00
0.63 -0.66 0.03 0.68 0.95 0.55 1.00
-0.31 0.24 -0.09 -0.90 -0.68 0.07 -0.56 1.00
0.18 -0.09 -0.25 -0.03 0.07 0.53 0.15 0.15 1.00
-0.15 0.37 0.00 -1.00 -0.74 -0.25 -0.67 0.92 0.05 1.00
0.75 0.05 0.16 0.13 0.30 -0.46 0.19 -0.51 -0.07 -0.18 1.00
0.08 0.43 0.16 -0.04 -0.28 -0.82 -0.44 -0.35 -0.28 -0.01 0.71 1.00
0.33 0.49 0.37 -0.61 -0.32 -0.61 -0.39 0.29 -0.24 0.58 0.50 0.50 1.00
Data for a second set of 82 polycyclic aromatic hydrocarbons are included in Table 5 while the correla-
tions are given in Table 6.
Table 5. Data for polycyclic aromatic hydrocarbons (PAH): melting point MP, boiling point BP, partition coefficient n-octanol/water Log P and the corresponding LEL, Wiener W and Randic X indices
No.	Molecule	MP	BP	LogP	LEL	W	X
1	naphtalene	81	218	3.35	13.341	109	4.966
2	1-methylnaphthalene	-22	245	3.87	14.572	140	5.377
3	2-methylnaphthalene	35	241	4	14.575	144	5.36
4	1-ethylnaphthalene	-14	259	4.39	15.837	182	5.915
5	2-ethylnaphthalene	-7	258	4.38	15.841	190	5.898
6	2-6-dimethylnaphthalene	110	262	4.31	15.808	186	5.754
7	2-7-dimethylnaphthalene	97	262	-	15.808	185	5.754
8	1-7-dimethylnaphthalene	-14	263	4.44	15.805	180	5.771
9	1-5-dimethylnaphthalene	80	269	4.31	15.802	176	5.788
10	1-2-dimethylnaphthalene	-4	271	4.31	15.803	178	5.788
11	1-3-7-trimethylnaphthalene	14	280	-	17.037	226	6.165
12	2-3-5-trimethylnaphthalene	25	285	-	17.035	224	6.182
13	2-3-6-trimethylnaphthalene	101	286	4.73	17.038	230	6.165
14	phenalene	85	-	-	17.919	210	6.449
15	1-phenylnaphthalene	45	334	-	21.739	412	7.949
16	2-phenylnaphthalene	104	360	-	21.744	436	7.933
17	anthracene	216	340	4.5	19.197	279	6.933
No.	Molecule	MP	BP	LogP	LEL	W	X
18	1-methylanthracene	86	363	-	20.426	334	7.343
19	2-methylanthracene	209	-	-	20.429	342	7.327
20	2-7-dimethylanthracene	241	370	-	21.66	413	7.72
21	2-6-dimethylanthracene	250	370	-	21.66	414	7.72
22	2-3-dimethylanthracene	252	-	-	21.658	408	7.737
23	9-10-dimethylanthracene	183	-	5.69	21.646	378	7.788
24	phenanthrene	101	338	4.52	19.194	271	6.949
25	1-methylphenanthrene	123	359	5.08	20.422	326	7.36
26	2-methylphenanthrene	56	355	5.24	20.425	334	7.343
27	3-methylphenanthrene	65	352	5.15	20.425	330	7.343
28	4-methylphenanthrene	50	-	-	20.422	322	7.36
29	9-methylphenanthrene	91	355	-	20.421	322	7.36
30	3-6-dimethylphenanthrene	141	363	-	21.656	396	7.737
31	4-5-methylenephenanthrene	116	359	-	21.195	300	7.433
32	tetracene	257	-	5.76	25.047	569	8.899
33	benzo [ a] anthracene	162	-	5.91	25.043	553	8.916
34	chrysene	256	441	5.86	25.039	545	8.933
35	benzo[c]phenanthrene	68	-	-	25.038	529	8.933
36	triphenylene	199	439	5.49	25.032	513	8.949
37	pyrene	156	393	5	22.49	362	7.933
38	1-methylpyrene	70	410	-	23.717	428	8.343
39	2-methylpyrene	144	410	-	23.72	434	8.327
40	4-methylpyrene	148	410	-	23.717	424	8.343
41	2-7-dimethylpyrene	-	396	-	24.949	515	8.72
42	pentacene	271	-	-	30.894	1011	10.865
43	dibenzo [ai] anthracene	264	-	6.81	30.889	987	10.882
44	dibenzo[ah] anthracene	270	-	5.8	30.885	971	10.899
45	dibenzo[aj]anthracene	198	-	-	30.885	955	10.899
46	benzo[b]chrysene	294	-	-	30.885	971	10.899
47	dibenzo[ac] anthracene	205	-	-	30.877	907	10.916
48	pycene	-	519	-	30.881	963	10.916
49	benzo[a]pyrene	177	496	5.97	28.331	680	9.916
50	benzo[e]pyrene	179	493	-	28.325	652	9.933
51	perylene	278	-	6.25	28.326	654	9.933
52	coronene	360	-	6.5	34.906	1002	11.899
53	anthranthrene	261	-	-	31.621	839	10.899
54	benzo [ghi]perylene	283	-	6.9	31.617	815	10.916
55	dibenzo[ae]pyrene	234	-	-	34.163	1082	11.916
56	1-methylchrysene	161	-	-	26.265	620	9.343
57	6-methylchrysene	257	-	-	26.267	632	9.343
58	3-methylcholanthrene	180	-	6.75	29.54	804	10.327
59	indeno[1-2-3-cd]pyrene	163	-	-	31.599	845	10.916
60	pentaphene	263	-	-	30.889	979	10.882
61	hexaphene	308	-	-	36.734	1589	12.848
62	indano	-51	178	-	12.043	79	4.466
63	indene	-2	183	2.92	12.043	79	4.466
64	azulene	100	270	3.22	13.335	107	4.966
65	acenaphthene	96	279	3.92	16.624	166	5.949
66	acenaphthylene	93	270		16.624	166	5.949
67	fluorene	117	294	4.18	17.899	219	6.449
68	1-methylfluorene	87	318	4.97	19.128	267	6.86
69	2-methylfluorene	104	318	-	19.131	274	6.843
70	3-methylfluorene	88	316	-	19.131	272	6.843
71	4-methylfluorene	71	-	-	19.128	265	6.86
72	9-methylfluorene	47	-	-	19.125	262	6.877
73	1-2-benzofluorene	190	407	5.4	23.746	461	8.433
74	fluoranthene	111	383	5.2	22.466	364	7.949
75	2-3-benzofluorene	209	402	5.75	23.75	471	8.416
76	3-4-benzofluorene	125	406	-	23.745	453	8.433
77	benzo [ghi] fluoranthene	149	432	5.78	25.759	478	8.933
No.	Molecule	MP	BP	LogP LEL	W	x
78	benzo [k] fluoranthene	217	481	- 28.313	698	9.916
79	benzo[b]fluoranthene	168	481	- 28.307	676	9.933
80	benzo [j ] fluoranthene	166	480	- 28.309	678	9.933
81	ovalene	473	-	- 47.307	2106	15.865
82	quaterryllene	483	-	- 58.242	4544	19.865
Table 6. Correlation of PAH properties with selected topological indices
Property	LEL	W	x
MP (n = 80)	0.857	0.748	0.855
BP (n = 53)	0.989	0.955	0.988
LogP (n = 37)	0.945	0.905	0.948
3. Results and Discussion
Wiener index is perhaps the most studied topological descriptor. It was used to predict the thermodynamic properties of hydrocarbons since the pioneering works of Wiener. The pool of Wiener-type descriptors is still in growth.
Other indices gained considerable attention, and revisited from time to time, to add new and important results. Among these, the Zagreb indices,31 the Hosoya indices and polynomials32 and the Cluj indices and polyno-mials33 are the best known.
In the present work we selected twelve TIs, all but LEL being extensions of the Wiener index. Thirteen pro-perties34,35 were considered for correlating study, their values in octane isomers and the correlation with the best scored TIs (among more than eighth hundred TIs provided by TOPOCLUJ software package) being listed in Table 1. Other results will refer to a set of aromatic hydrocarbons (see below).
Table 2 contains properties from the benchmark data sets of Milano Chemometrics & QSAR Research Group34, except HV (Heat of Vaporization) - which is taken from the Korea Thermophysical Properties Data Bank35. Nonanes and decanes data are taken from the Korea Thermophysical Properties Data Bank, while polyaro-matic hydrocarbons (PAH) are taken from the Milano Chemometrics & QSAR Research Group. All correlations were made in monovariate linear regression, to have a more direct interpretation of the results and a good insight on the structure-property relationship.
Boiling point BP, one of the difficult to model properties, is the best correlated by the newly proposed descriptor LEL: R = 0.673. In n-alkanes, up to eicosane, the Pearson correlation index is R = 0.978. The index clearly bears size information because in the set of nonanes R = 0.525 while in decane isomers R = 0.265. The second well correlated among the selected indices, in the set of octanes, is 1/1WK, (0.599). Compare with the classical Wiener index for which R = 0.559.
Octane number MON, related to the motor combustion, is the best modeled by 2WD, a Wiener-type index of rank 2 (-0.966), LEL being this time second from the bottom but still highly correlated (-0.941).
Heat of vaporization HV, is the best modeled by LEL (0.887), next being 1/1WK, (0.843).
The three above properties are the worst described by PDS3 [Sh[SZ]] (0.432, 0.561, -0.010, respectively). This index, calculated on a shell-type matrix is practically uncorrelated with all the other indices herein discussed and fit the best in case of "recalcitrant" properties, when the other indices failed (see Table 2, last row, the italicized entries). The matrix of index inter-correlation is given in Table 3. It can be seen that all but the last are related higher than 0.94. LEL is well correlated with the WK indices.
Molecular volume MV, molecular refraction MR and density DENS are highly correlated properties (see Table 4). These are modestly described by the all indices except PDS3 [Sh[SZ]], as above mentioned.
Acentric function AF, is the property on which we focus the attention in the following. This parameter is a well described one by all indices excepting the last index. The best score was recorded by 1WK (-0.995); the newly introduced index LEL is also highly correlated to this parameter in heptanes (0.987), octanes (0.990), nonanes35 (0.978) and decanes35 (0.975).
A second set of aromatic hydrocarbons34 (Table 5) was also investigated: melting point MP, boiling point BP and the partition coefficient n-octanol/water LogP, vs. three TIs, LEL, Wiener W and Randic x index.36 The results show LEL as good as x index and better than Wiener index (Table 6). Within this set of molecules, the degeneracy is comparable for the three indices; LEL is correlated here with the Wiener index W by 0.914 while with the Randic index x by 0.999.
4. Conclusions
A correlating study of topological indices provided by TOPOCLUJ software package and LEL, a newly proposed index, on thirteen properties of octanes revealed good correlating ability of a dozen selected TIs, all related to the Wiener index.
LEL is the best correlated with the WK indices. It describes well the properties which are well accounted by the majority of the selected molecular descriptors: octane number MON, entropy S, volume MV, or refraction MR,
particularly the AF parameter, but also more difficult properties like boiling point, melting point and logP. Among the desirable attributes required by a good TI, LEL fulfils: good correlation with at least one property, good discrimination of isomers and simplicity. In addition, it is well defined mathematically and shows interesting relations in particular classes of graphs. This index and those provided by the TOPOCLUJ software as well, was proved to be of basic importance in QSAR/QSPR studies.
5. Acknowledgement
This work is supported in part by the research program P1-0285 of the Slovenian Agency for Research and the grants 144015G and 144007 of the Serbian Ministry of Science and Technological Development and in part by the Romanian GRANT ID_506.
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Povzetek
Deskriptorje dobljene s programskim paketom TOPOCLUJ in indeksom LEL osnovanim na Laplaceovi matriki smo uporabili za izgradnjo korelacijskih modelov, ki služijo za opis trinajstih različnih lastnosti oktanov. Deskriptor LEL zelo dobro opiše večino izbranih lastnosti, kot so: oktansko število, entropija, volumen molekule, indeks refrakcije in parameter AF. Deskriptor LEL je uporaben tudi za opis bolj kompleksnih lastnosti kot sta temperaturi vrelišča in tališča, ter logP. Opazili smo dobro korelacijo indeksa LEL z indeksi WK, izračunanimi s programom TOPOCLUJ. Ob ko-relacijski analizi drugega obravnavanega seta spojin - policiklični aromatski sistemi se je LEL izkazal tako dobro kot Randicev X indeks in bolje kot Wienerjev indeks. Naša študija je pokazala, da je novi topološki indeks LEL uporaben za matematični opis določenih razredov grafov.