Real number vertex invariants: Regressive distance sums and related

Ovidiu Ivanciuc, Teodora Ivanciuc, Douglas J. Klein, William A. Seitz, and Alexandru T. Balaban. Journal of Chemical Information and Computer Sciences...
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421

J. Chem. If: Comput. Sci. 1993, 33, 421-428

Real Number Vertex Invariants: Regressive Distance Sums and Related Topological Indices Alexandru T. Balaban' Department of Organic Chemistry, Polytechnic University, Splaiul Independentei 3 13, 77206 Bucharest, Romania Mircea V. Diudea Department of Organic Chemistry, "Babes-Bolyai" University, Strada Arany Janos 11, 3400 Cluj, Romania Received September 18, 1992 A new type of layer matrix, called R matrix, was constructed on the basis of distance sums of vertices. This matrix was operated with two classes of operators: one of "centricity" ("c") type and the other of "centrocomplexity" ("x") type, the last one taking into account the "more important" vertices in molecular graphs. The matrix invariants are computed with a TURBO PASCAL, TOPIND 10 program for various examples, and finally an intercorrelating matrix is given for the proposed topological indices in the set of heptane isomers. INTRODUCTION Various papersl-14advocated the use of real number vertex invariants for obtaining less degenerate topological indices (TIS). First-generationTIs had been integer numbers obtained on the basis of integer-number local vertex invariants (LOVIs) (e.g. the Wiener indexI5). Second-generation TIS were real numbers obtained from integer LOVIs using more sophisticated operations; such TIS are RandiC's molecular connectivity,I6 Kier and Hall's extended molecular c~nnectivity,'~ all information-theoretic and the average distance sum connectivity, J.21 The newest (third-generation) TIS are real numbers based on real-number LOVIs. Among these ones, in a recent paper,22 we introduced new LOVIs called regressive vertex degrees. We started from the idea that TISbased on the distance matrix of graphs reflect the more distant relationships between graph

4

ri

4 5

ri*

i

110 2 71 3 61 4 71 510

7 6 71 6 710 420 0 6 710 7 6 71

0 0 0 0 0

for k = 0-2

fork = 0-3 0 1 2 3

0 1

8 51 522 6141 9 6 8 51

7 421 428 0 7 421 7 421 7 421

4 9 9 0 6 0 516 4 9

2

Figure 1. R matrix in graphs GI-(&.

0 1 6 5 5 6 6

1 1 1 1 1

2

012 7 6 7 6 111 111

J = 2.5395

10.070 607 1 7.160 7100 6.142 0000 7.160 7200 10.070 607 1

8.051 409 5.220 900 6.141 600 9.060 5 16 8.051 409

7.0421 4.2800 7.0421 7.0421 7.0421

6.1012 5,1706 5.1706 6,1111 6.1111

J = 3.0237

1 2 3 4 5

0.099 30 0.139 65 0.162 81 0.139 65 0.099 30

0.124 20 0.191 54 0.162 82 0.1 10 37 0.124 20

0.142 00 0.233 64 0.142 00 0.142 00 0.142 00

R+

J = 2.1939

0.640 7 1

0.713 13

0.801 66

1 2 3 4 5 RC

0.143 88 0.208 74 0.320 25 0.208 74 0.143 88

0.208 20 0.343 17 0.328 61 0.205 20 0.208 20

0.334 77 0.716 61 0.334 77 0.334 77 0.334 77

0.344 51 0.362 52 0.362 52 0.346 46 0.346 46

1.025 49

1.293 38

2.055 70

1.762 45

rxi

1 2 3 4 5

0.099 30 0.279 30 0.325 63 0.279 30 0.099 30

0.124 20 0.574 61 0.325 65 0.110 37 0.124 20

0.142 00 0.934 58 0.142 00 0.142 00 0.142 00

0.327 80 0.580 20 0.580 20 0.327 27 0.327 27

h!x

1.082 83

1.259 03

1.502 59

2.142 76

rji

1 2 3 4 5 RJ

0.117 76 0.268 55 0.301 58 0.268 55 0.117 76 1.074 19

0.154 24 0.485 07 0.310 65 0.134 05 0.154 24 1.238 26

0.182 15 0.728 60 0.182 15 0.182 15 0.182 15 1.457 19

0.356 08 0.549 34 0.549 34 0.341 53 0.341 53 2.137 83

dji

1 2 3 4 5 DJ

0.119 52 0.273 83 0.308 61 0.273 83 0.119 52 1.095 30

0.158 11 0.498 80 0.318 66 0.136 08 0.158 11 1.269 77

0.188 98 0.755 93 0.188 98 0.188 98 0.188 98 1.511 86

0.365 15 0.565 15 0.565 15 0.349 24 0.349 24 2.193 93

I

=&

for k = 0-4 o 1 2 3 4

J = 2.1906

rci

6

Ol*OI

64

0.163 90 0.193 40 0.193 40 0.163 64 0.163 64 0.877 98

Distance sums

.

1

2 3

G3 @)

G3

61

I

(a) Numbering of Vertices

.e:

Figure 2. Local and global invariants based on R matrix in GI+.

0095-2338 /93 / 1633-0421SQ4.QOlQ , 0 1993 American Chemical Society ,

I

422 J. Chem. In& Comput. Sci., VoI. 33, No. 3, 1993

BALABANAND DIUDEA

65

(a) Actual Parameters no. 3 no. 4

rl

r,*

reI

rx,

rj,

dj,

17.063 104 062 17.063 108 058

0.058 605 984 0.058 605 970

0.133 44 0.13432

0.175 817 95 0.175 817 91

0.15974 0.159 46

0.16026 0.15997

(b) Other Parameters, Including the Eccentricity,ecc, ecc,

no. 3 no. 4

frkZ2

3 3

3,4,2 3,4,2

bik22

self returning walks,*’

random walksz3

3,7,6,2 3,7,6,2

0,3,0,13,0,59,0 0,3,0,13,0,61,0

3,7, 15, 33,73,159 3,7,15, 35, 73, 169

Figure 3. Vertex discrimination in 234MMEC6,Gs.

vertices. In this respect, the distance sums for each vertex are dominated by the more remote vertices; on the other hand, most TIS based on the adjacency matrix emphasize only the immediate vicinity relationships. By means of the regressive vertex degrees, we extended the neighborhood relationship to also include more distant vertices but in attenuated form, their contribution decreasing with increasing distance.22 In the present paper we perform a similar operation for distance sums, in order to convert them into real numbers. The new LOVIs herein proposed may include information about multiple bonding and about heteroatoms, as will be shown below, REGRESSIVE DISTANCE SUMS One starts by calculating the distance sums D;for all vertices in graph G, i.e. by summing entries over rows or columns in the distance matrix (Di = Z,d;j). Next, one writes a new matrix which will be called the R matrix (for regressive distance sums) according to the various shells around each vertex i: the entry in column k = 0 is just the distance sum, D;. The next columns will sum all distance sums, D,, of vertices j , belonging to a shell at distance d;, = k, around the vertex i. Thus, the entries in R matrix will be

10

I

+ ’% + + 11l

G6 rei, bei ri* rxi, rji dji

bxi

5,4,6,3, 7, 2,8, 1 = 10 = 11,9 4,5,3,6,2,7,1 = 10 = 11, 8,9 2,4,5, 3, 6,7,8, 1 = 10 = 11,9 2,4,5 = 3,6,7,8, 1 = 10 = 11,9 2,3,4,5, 6,7,8, 1 = 10 = 11,9

Figure 4. Vertex ordering in 22MMC9,G6.

identification in G, so that it can be used for centric ordering ofverticcs(seerefs24and25). Sincether;parametersbecome cumbersome in large graphs, for an easier handling of the R matrix we propose four operators, defining four other LOVIs, whose first letter is r, as follows: (3)

(4) k=l

rx; = [ri/dgj - mj]-’wi The number of columns in R is equal to the largest distance in G, (Le. the graph diameter). It is obvious that the sums over each row in R are all equal to twice the Wiener index (the sum of all distances in G). We exemplify on four graphs with five vertices, as in Figure 1. By analogy with the regressive degrees22(which count the decreasingcontributions of more remotevertices to the classical degree of a vertex, as their distance to that vertex increases), we propose new real-number LOVIs, regressivedistance sums, defined as diam

where diam is the diameter and n denotes the number of digits for the maximal r i k value in G. This vertex invariant is directly related to the second criterion (0;= min), established by Bonchev et al.23for the center

mi =f;.[rio/10

+ ril/lOO]

(5)

(6)

(7)

ci = 1 +A.

(9)

where dSF is a specified distance value, usually larger than the largest path in G24v25(in the following, dswc= 10 unless otherwise specified); mi is the local parameter for multiple bonds; ci, f;. refer to the connectivity around the vertex i; c;, is the conventional bond order, 1, 2, 3, and 1.5 for single, double, triple, and aromatic bonds, respectively; and w; is a weighting factor, accounting for heteroatoms, as defined in refs 22 and 26. Summation of the new LOVIs over all ivertices in G provides the corresponding global indices (TIS), denoted by capital

J. Chem. Inf. Comput. Sci., Vol. 33, No. 3, 1993 423

REALNUMBER VERTEXINVARIANTS

5-1 G11 Jhei rxi

2 3 1 5 4

RX

J = 2.4017

67 2.5030

rxi dji

Til

0.73445 0.45028 0.201 47 0.201 47 0.18403 1.771 70

2 3 1 5 4 RJ

0.678 94 0.438 32 0.22209 0.222 09 0.203 55

2 3 1 5 4

1.76499 DJ

2 3 1 5 4

1.809 36

BX

4.358 51 3.346 21 2.159 01 2.159 01 2.054 27

2A

2 3 4 1 5

0.448 76 0.36403 0.268 55 0.235 52 0.117 76

2 3 4 1 5

0.457 26 0.372 52 0.273 83 0.23905 0.119 52

RX

1.18293

RJ

1.43462

DJ

1.462 18 BX

rxI

djl

bxi

I

0.73445 0.45028 0.207 10 0.201 47 0.17903

2 3 1 5 4

0.68203 0.435 54 0.225 17 0.22209 0.200 77

2 3 1 5 4

0.701 21 0.446 52 0.23083 0.227 67 0.203 80

2 3 1 5 4

W

1.77232

RJ

1.765 58

DJ

1.81003

BX

! I i

2.551 00 2.42000 2.321 00 1.342 10 1.222 10 9.85620

612 J = 2.6224

b

rjl

2 3 4 1 5

2

14.07700

4.358 5 3.346 2 2.219 34 2.159 0 1.998 4:

rji

bxi

dji

5

0.429 00 0.367 60 0.279 30 0.099 30 0.099 30

3 2 4 1 5

0.51482 0.468 11 0.331 01 0.16654 0.11776

3 2 4 1 5

0.52682 0.477 64 0.337 74 0.16903 0.11952

3 2 4 1 5

2.660 OC 2.551 OC 2.321 OC 1.222 1c 1.222 1c

Ry

1.274 50

RJ

1.59824

DJ

1.63076

BX

9.976 2C

3 rxi

bxi

0.367 60 0.325 63 0.279 30 0.111 10 0.099 30

l5

N'

dji

2 3 4 1 5

bxi

0.698 05 0.449 35 0.227 67 0.227 67 0.206 63

rji

2 3 I

'T

14.081 4'

G13 J = 2.8257 rji

?XI

4,

rj,

rxi

bxi

I

0.73445 0.463 23 0.201 47 0.201 47 0.17903

2 3 1 5 4

0.682 30 0.441 76 0.22209 0.222 09 0.20363

2 3 1 5 4

0.701 52 0.45289 0.227 67 0.227 67 0.20671

2 3 1 5 4

W

1.77964

RJ

1.771 86

DJ

1.81646

BX

! I i

4.358 5 3.44241 2.159 0 2.159 0 1.998 4:

3 1 I 5

bxi

dji

I

0.574 61 0.429 04 0.124 20 0.124 20 0.123 45

3 2 1 4 5

0.558 22 0.517 86 0.268 11 0.15424 0.15424

3 2 1 5 4

0.57443 0.53036 0.272 17 0.158 11 0.158 11

3 2 1 4 5

Rx

1.375 50

RJ

1.652 67

DJ

1.693 18

BX

3.410 O( 2.660 O( 1.352 O( 1.331 O( 1.331 O( 10.084 O(

'T

14.1173!

Gl4 J = 2.8474 rx,

3

I

0.768 17 0.45028 0.201 47 0.201 47 0.17903

W

1.80042

! I i

rji

2 3 1

bx,

dji

4

0.694 36 0.44087 0.227 13 0.227 13 0.200 77

2 3 1 5 4

0.713 90 0.45203 0.23284 0.23284 0.203 80

2 3 1 5 4

RJ

1.79025

DJ

1.83540

BX

5

4.558 6' 3.346 2 2.1590 2.1590 1.998 4:

I

0.776 35 0.383 80 0.30848 0.218 13 0.13405

3 2 4 5 1

0.798 03 0.39428 0.31623 0.22361 0.13608

3 2 4 5 1

Rx

1.679 18

RJ

1.82081

DJ

1.86823

BX

5

[ ~ i / ( ~ i c i ) ~ j / ( ~ , ~ j ) ~ - " *

10.194 O(

G15 J = 3.1943

letters, corresponding to the respective LOVIs. If ri is replaced by Di in eq 8 , a new vertex invariant can be designed:

C

3.750 O( 2.420 O( 1.461 O( 1.331 O( 1.232 O(

'Y

14.221 3:

Figure 5. Heteroatom perception.

djt =

bxi

dji

3 2 4 5 1

1 I rxi

rji

0.980 10 0.325 65 0.138 86 0.124 20 0.11037

(10)

(id

It is easily seen that, when wi = 1 and Ci = 1, the corresponding global index, DJ, is related to the J index21

rx, 3 1 I 5 I Rx

0.980 10 0.429 04 0.124 20 0.124 20 0.1 10 37 1.767 91

rji

3 2 4 5 1 RJ

0.789 45 0.542 78 0.218 13 0.218 13 0.18958 1.958 06

bxi

dji

3 2 4 5 1

DJ

Figure 6. Double bonding perception.

0.812 36 0.557 60 0.223 61 0.22361 0.19245 2.00962

3 2 4 5 1

BX

3.750 O( 2.660 O( 1.331 O( 1.331 O( 1.232 O( 10.304 O(

424 J . Chem. Inf. Comput. Sci., Vol. 33, No. 3, 1993

BALABANAND DIUDEA

Table I. R Matrices and RC Index, for G16 and GI,, According to Their Centric Numbering (dSw = 20)O G16: 42 Vertices and 51 Edges 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

161 161 169 169 173 173 181 181 181 181 197 197 205 205 209 209 209 209 225 225 233 233 237 237 24 1 24 1 245 245 26 1 26 1 273 273 277 277 28 1 28 1 297 297 309 309 313 313

507 499 539 539 547 547 559 559 563 563 394 394 406 406 418 418 667 667 695 695 446 446 707 707 719 719 490 490 522 522 546 546 554 554 558 558 570 570 570 570 558 558

RC 1 2 3 4 5 6

0.0273258 0.027 3198 0.021 2683 0.021 2683 0.0207258 0.0207258

1110 1102 945 945 951 957 977 977 1230 1230 840 840 583 583 848 848 1145 1145 948 948 659 659 948 948 944 944 735 735 743 743 73 1 731 723 723 486 486 498 498 498 498 486 486

1640 1640 1431 1431 1419 1419 1519 1519 1559 1559 1290 1290 1001 1001 1310 1310 1057 1057 65 1 65 1 900 900 647 647 635 635 864 864 651 65 1 639 639 63 1 631 422 422 434 434 434 434 442 442

RC 7

8 9 10 11 12

0.0154987 0.0154987 0.0154075 0.0154075 0.0149788 0.0149788

1840 1880 208 1 208 1 1812 1812 1912 1912 1366 1366 1242 1242 1491 1491 1318 1318 768 768 51 1 511 796 796 543 543 539 539 744 744 342 342 350 350 354 354 587 587 378 378 378 378 386 386

21 12 21 12 2040 2040 1731 1731 1430 1430 1390 1390 1511 1511 1535 1535 1342 1342 1053 1053 933 933 933 933 1021 1021 768 768 768 768 511 511 543 543 539 539 744 744 342 342 3 50 350 354 354

RC 13 14 15 16 17 18

1766 1734 1369 1369 1626 1626 940 940 1217 1217 1731 1731 1418 1418 1133 1133 1390 1390 1511 1511 1254 1254 1342 1342 1053 1053 1053 1053 933 933 1021 1021 768 768 768 768 511 511 543 543 539 539

618 626 867 867 1180 1180 1056 1056 1056 1056 1369 1369 1626 1626 940 940 1217 1217 1731 1731 1418 1418 1133 1133 1390 1390 1390 1390 1511 1511 1342 1342 1053 1053 1053 1053 933 933 1021 1021 768 768

RC

0.0145172 0.0145172 0.011 0351 0.011 0351 0.0109346 0.0109346

19 20 21 22 23 24

0.0106084 0.0106084 0.0102561 0.0102561 0.0078730 0.0078730

313 313 309 309 867 867 883 883 867 867 1180 1180 1056 1056 1056 1056 1369 1369 1626 1626 940 940 1217 1217 1217 1217 1731 1731 1133 1133 1390 1390 1390 1390 1511 1511 1342 1342 1053 1053

313 313 309 309 313 313 309 309 867 867 883 883 867 867 1180 1180 1056 1056 1056 1056 1056 1056 1369 1369 940 940 1217 1217 1217 1217 1731 1731 1133 1133 1390 1390

313 313 309 309 313 313 309 309 867 867 883 883

313 313 309 309

883 867 867 1056 1056 1056 1056 1056 1056 1369 1369 940 940 1217 1217

309 313 313 867 867 883 883 883 883 867 867 1056 1056 1056 1056

RC 25 26 27 28 29 30

0.0077722 0.0077722 0.0077552 0.0077552 0.007 5071 0.0075071

313 313 309 309 309 309 313 313 867 867 883 883

RC 31 32 33 34 35 36

0.0056155 0.0056155 0.0055346 0.005 5346 0.0055187 0.0055187

RC 37 38 39 40 41 42

global index, RC: 0.466 826 603 GI,: 42 Vertices and 51 Edges 1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

161 161 169 169 173 173 181 181 181 181 197 197 205 205 209 209 209 209 225 225 233 233 237

503 503 539 539 547 547 559 559 563 563 394 394 406 406 418 418 667 667 695 695 446 446 707

1106 1106 949 949 953 953 977 977 1230 1230 840 840 583 583 848 848 1145 1145 948 948 659 659 948

1640 1640 1435 1435 1415 1415 1523 1523 1555 1555 1294 1249 997 997 1310 1310 1057 1057 651 65 1 900 900 647

1860 1860 2081 2081 1812 1812 1916 1916 1362 1362 1246 4 1487 1487 1322 1322 764 764 515 515 792 792 543

2112 2112 2020 2020 1751 1751 1430 1430 1390 1390 1511 1246 1535 1535 1346 1346 1049 1049 937 937 929 929 1025

1750 1750 1369 1369 1626 1626 920 920 1237 1237 1711 1511 1438 1438 1133 1133 1390 1390 1511 1511 1254 1254 1346

622 622 883 883 1164 1164 1056 1056 1056 1056 1369 1711 1626 1626 920 920 1237 1237 1711 1711 1438 1438 1133

309 309 313 313 883 883 867 867 883 1369 1164 1164 1056 1056 1056 1056 1369 1369 1626 1626 920

309 309 313 313 309 883 313 313 883 883 867 867 883 883 1164 1164 1056

309 309 309 313 313 309 309 313 313 883

313 313 309 309

309

0.0053242 0.0053242 0.0040122 0.0040122 0.0039474 0.0039474

REALNUMBER VERTEXINVARIANTS

J. Chem. Inf. Comput. Sci., Vol. 33, No. 3, 1993 425

Table I (Continued) G17: 42 Vertices and 51 Edges

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

237 241 241 245 245 261 261 273 273 277 277 281 281 297 297 309 309 313 313

707 719 719 490 490 522 522 546 546 554 554 558 558 570 570 570 570 558 558

RC 0.0273227 0.0273227 0.021 2669 0.021 2669 0.0207212 0.0207212

1 2 3 4 5 6

948 944 944 73q5 735 743 743 731 731 723 723 486 486 498 498 498 498 486 486

647 635 635 864 864 651 651 639 639 631 631 422 422 434 434 434 434 442 442

543 539 539 744 744 342 342 350 350 354 354 587 587 318 378 378 378 386 386

RC 7 8 9 10 11 12

1025 764 764 764 764 515 515 543 543 539 539 744 744 342 342 350 350 354 354

1346 1049 1049 1049 1049 937 937 1025 1025 764 764 764 764 515 515 543 543 539 539

RC

0.0155020 0.0155020 0.0154065 0.0154065 0.0149777 0.0149777

13 14 15 16 17 18

1133 1390 1390 1390 1390 1511 1511 1346 1346 1049 1049 1049 1049 937 937 1025 1025 764 764

920 1237 1237 1237 1237 1711 1711 1133 1133 1390 1390 1390 1390 1511 1511 1346 1346 1049 1049

RC

0.0145135 0.0145135 0.011 0369 0.011 0369 0.0109347 0.0109347

19 20 21 22 23 24

0.0106066 0.0106066 0.0102542 0.0102542 0.0078739 0.0078739

1056 1056 1056 1056 1056 1369 1369 920 920 1237 1237 1237 1237 1711 1711 1133 1133 1390 1390

883 867 867 867 867 883 883 1056 1056 1056 1056 1056 1056 1369 1369 920 920 1237 1237

RC 25 26 27 28 29 30

309 313 313 313 313 309 309 883 883 867 867 867 867 883 883 1056 1056 1056 1056

309 309 313 313 313 313 309 309 883 883 867 867

RC

0.0077727 0.0077727 0.007 7556 0.0077556 0.0075053 0.007 5053

31 32 33 34 35 36

309 309 313 313

RC

0.0056159 0.0056159 0.005 5353 0.005 5353 0.005 5194 0.005 5194

37 38 39 40 41 42

0.0053226 0.005 3226 0.0040122 0.0040122 0.0039481 0.0039481

global index, R C 0.466 807 649 (1

Missing entries in shorter columns are zeros.

A 4f

42

35

36

40

44 35

36

42

38 37

G16

7

Figure 7. Centric numbering of graphs G16 and GI,.*’

as follows: DJ = 2J(p

+ l)/q

(1 1) where p is the cyclomatic number and q is the number of edges in G. The above LOVIs and TIS are examplified for GI-G~ in Figure 2. INTRAMOLECULAR ORDERING That the R matrix is more powerful in discriminating nonequivalent vertices than FZ4and BZ2matrices and their derived invariants or other topological descriptors can be seen in Figure 3 for G5.23 (i) %”-Typeversus “x”-TypeOrdering. Our invariants are capable of ordering the vertices in molecular graphs either in terms of centricity (“c”) or centrocomplexity( “ x ” ) . * ~Figure

4 shows the ordering given by operators rci, ri*, m i , rji, and dji and the older operators bci and bxj (the last one denoted as BU2)in ref 22 and as BCX in ref 25). (ii)Heteroatoms and Multiple Bonding. The operators mi, rji, djj, and bxi are sensitive to the presence of heteroatoms and multiple bondings by means of the Wi and mi (or ci) factors.

Weexemplifythis with a set of amines and alkenes, the LOVIs values being given in Figures 5 and 6, in decreasing order. One can see that our invariants (the earlier BX included) emphasizethe centricity of heteroatomsand multiple bonding, their values paralleling the centricity of ”important” vertices in graphs. INTERMOLECULAR ORDERING OF ISOMERIC GRAPHS The R matrix surpasses the ability of the known layer matrices F and B to discriminate between isomeric graphs.

BALABANAND DIUDEA

426 J. Chem. InJ Comput. Sci., Vol. 33, No. 3, 1993 (a) New TIS G

isomer

R*

RC

Rx

RJ

DJ

18 19 20 21 22 23 24 25 26

Cl 2MC6 3MC6 24MMCs 3EC5 23MMC5 22MMC5 33MMC5 223MMC.j

0.4490 0.4825 0.5040 0.5212 0.5267 0.5464 0.5461 0.5728 0.5959

0.6412 0.7802 0.8303 1.0452 1.0782 1.1103 1.0770 1.1931 1.4062

0.8035 0.8189 0.9306 0.9692 0.9849 1.0339 1.0343 1.1043 1.1631

0.8OOO 0.8730 0.9223 0.9599 0.9740 1.0208 1.0211 1.0865 1.1422

0.8158 0.8928 0.9439 0.9844 0.9974 1.0481 1.0515 1.1201 1.1804

(b) Previous TIS and van der Waals Areas, A G

isomer

BQ5

18 19 20 21 22 23 24 25 26

Cl 2MC6 3MC6 24MMC5 3EC5 23MMC5 22MMCs 33MMCs 223MMC4

1.324 95 1.472 44 1.540 56 1.737 82 1.802 68 1.838 00 1.787 26 1.938 40 2.148 56

BxZ5 14.395 06 14.615 04 14.636 82 14.836 80 14.658 60 14.876 40 15.054 60 15.094 20 15.312 00

(A2)

728

23.6523 24.5823 25.0396 25.5709 25.4757 26.2208 26.2438 26.8844 27.6039

NVZ9 1

112 297 364 56 1 1402 3546 5472 8508

~~(1130 13.4246 14.7656 15.0821 16.3631 15.3665 16.9492 17.9498 18.4853 20.5470

A2

334.36 322.91 316.67 309.97 303.15 303.1 1 306.53 297.20 292.89

(c) Intermolecular Ordering of Heptane Isomers GI&& RX, RJ, DJ, B(V: 18, 19, 20,21, 22, 23, 24, 25,26 R*: 18, 19, 20,21,22,24,23, 25,26 RC, BC: 18, 19, 20,21, 24, 22, 23,25, 26

BX, 7 , DM(1): 18, 19,20, 22, 21, 23, 24,25, 26

Figure 8. Global indices in heptane isomers (M = methyl; E = ethyl)

Dobrynin found a very interesting pair of catacondensed Their algorithm provides a set of 7-indices with high discriminating power which are useful in topological equivalence benzenoid graphs,27whose centricnumbering is given in Figure perception, and also in QSAR. 7. On comparing the intramolecular vertex ordering in These graphs show identical F and B,but not R,matrices. 22hfhfC9, Ga, one can see that c operators, rci and bci, order All TIS based on these matrices are identical, except the RC vertices alternatively vs central vertex no. 5. Notice that such index. It is obvious that the degeneracy of matrix invariants operators find the center of the graph according to the first induce the degeneracy of all derived TIS. criterion of Bonchev et al.23(minimal eccentricity). The two graphs show indeed different R matrices, but their Conversely, (an x operator), "sees" vertex no. 2 (with sums on columns are identical, so that the degeneracy of TIS degree 4) as the most important vertex in that graph, the appears at the operational stage (simple summation over all remaining vertices being ordered according to their increasing vertices in graph). Only a more sophisticated function, that distance from vertex no. 2. A quite similar ordering is given is the centric R C index, may discriminate between the two by operators rxi, rji, and dji (which, explicitly or implicitly, graphs. R matrices and the R C index for these graphs are all take into account the vertex degree). presented in Table I. The ri* index behaves differently, alternating the vertices relative to vertex no. 4 (considered as center, in agreement Global ordering in agreement with the c and x concepts, in with the second criterion of Bonchev et minimal distance the set of heptane isomers, is shown in Figure 8. sums). On considering Figure 8, it may be seen that in the set of Supplementary examples are given in Figure 9 for illusisomericheptanes the new TISlead to several distinct orderings. trating the equivalent vertex discriminating power of our It is interesting to observe that indices RX,RJ, and DJ give indices. Four cubic graphs are taken from ref 28 along with the same ordering as the "ideal" one advocated by B e r t ~ , ~ ~ the corresponding values (Sr = C,GMij/nij, as LOVIs which is identical with that induced by the J index.21 within r-index28). Despite different ranking of LOVIs, with one exception DISCUSSION (G29; operators R, RC,and RX),the equivalenceclasses were correctly found. The idea of "seeing" the total graph environment of each From the above examples, some remarks emerge: vertex/atom, developed by us in connection with layer matrices (i) c operators enhance the contribution of more remote B and R (see refs 22, 25, and 31 and this work) was also vertices, whereas x operators emphasize that of the nearer considered by Hall and KierZ8in constructingthe "topological neighbors. state" matrix and related 7-indices. They have defined the (ii) rci is the best c-type operator and 6xi is the best x-type overall structural relationships of a vertex i, making use of all one. Indices rx,, rji, and dji are not pure x operators since paths joining that vertex with each of the other vertices in G, they are based on Df, which is a c parameter. and the geometric mean, GMv, of +values for the chemical (iii) The ri* operator is a crude one and represents a natureof each vertexj belonging to a given path of nijvertices. compromise between c- and x-types.

J. Chem. Inf. Comput. Sci., Vol. 33, No. 3, 1993 427

REALNUMBER VERTEXINVARIANTS

10 7

6

(iv) Heteroatoms and multiple bondings are located mainly as “complexity” subgraphs. (v) An intercorreleting matrix (Figure 10) shows the RC index being a part of the x-type indices (7-index28included). The proposed indices correlate well with van der Waals areas2 in heptane isomers. (vi) Our indices are good tools in vertex equivalence perception; when the connectivity is taken into account (RJ

RC RC R

RX RJ DJ 728

A2

R*

RX

RJ

1.0000 0.9748 0.9732 0.9736 1.oooO 0.9993 0.9995 1.oooO 0.9999 1.oooO

DJ

T~~

0.9726 0.9992 0.9999 0.9999 1.oooO

0.9690 0.9969 0.9972 0.9974 0.9978 1.oooO

A2

0.9696 0.9717 0.9652 0.9664 0.9632 0.9527 1.m

BALABANAND DIUDEA

428 J . Chem. If. Comput. Sci., Vol. 33, No. 3, 1993

CONCLUSIONS The R matrix (layer of distance sums) represents an extension of F (layer of neighbors/distance frequencies) and B (layer of degrees) matrices and is suitable for topological index design. Its discriminating power surpasses that of the previous layer matrices. The intramolecular ordering of vertices, in c or x terms leads to real-number LOVIs, which can be used in QSAR/ QSPR studies (see ref 32). ACKNOWLEDGMENT Thanks are addressed to Dr. A. Dobrynin, Institute of Mathematics, Russian Academy of Sciences,Siberian Branch, Novosibirsk, for a preprint. We also thank Dr. D. Horvath, Department of Chemistry, “Babes-Bolyai”University, Cluj, Romania, for computer assistance. REFERENCES AND NOTES (1) Balaban, A.T.Usingrealnumbersasvertex invariantsforthirdgeneration topological indexes. J . Chem. InJ Comput. Sci. 1991, 32, 23-28. Balaban, A. T.; Balaban, T. S. Correlation using topological indexes based on real graph invariants. J. Chim. Phys. 1992,89, 1735-1745. (2) Labanowski, J.; Motoc, I.; Dammkoehler, R. A. The physical meaning of topological indices. Comput. Chem. 1991, 15, 47-53. (3) Rouvray, D. H. The limits of applicability of topological indices. J. Mol. Struct. (THEOCHEM) 1989,54, 187-201. (4) Ivanciuc, 0.;Balaban, T. S.; Balaban, A. T. Reciprocal distance matrix, !elated local vertex invariants and topological indices. J . Math. Chem., (5) (6) (7)

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