J. Chem. Inf. Comput. Sci. 1997, 37, 1095-1100
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Matrix Operator, W(M1,M2,M3), and Schultz-Type Indices Mircea V. Diudea*,† and Milan Randic´‡ Department of Chemistry, “Babes-Bolyai” University, 3400 Cluj, Romania, and Department of Mathematics, Drake University, Des Moines, Iowa 50311 Received April 1, 1997X
Recently proposed matrix operator, W(M1,M2,M3), is used to generate the molecular topological index, MTI, and novel Schultz-type descriptors. Matrix algebra involved by this operator is exemplified and discussed in comparison to the Cramer algebra. INTRODUCTION
Among the modifications of the Wiener index,1 W, the molecular topological index, MTI, proposed by Schultz,2 appears to be one of the most studied topological indices.3-18 It is defined as
MTI ) ∑[v(A + D)]i
(1)
i
Recall that the Wiener index, in acyclic structures, can be calculated1 by
W ) ∑NiNj
(6)
(i,j)
where Ni and Nj denote the number of vertices on the two sides of the edge (i,j). The products NiNj are just the entries in the Wiener matrix,22,23 We, so that W can be calculated by
where A and D are the adjacency and the distance matrices, respectively, and v ) (V1, V2, ..., VN) is the vector of the vertex valencies/degrees in a connected graph, G. Recall that the entries in A are equal to unity if vertices i and j are adjacent and zero otherwise. The distance matrix collects the number of edges on the shortest path joining the vertices i and j. By applying the matrix algebra, MTI can be written8,13,15,19,20 as
In cycle-containing graphs, W can be calculated by means of the D matrix24
MTI ) uA(A + D)uT ) S(A2) + S(AD)
Gutman15 has expressed the S(AD) index by analogy to the Wiener index (cf. eq 6)
(2)
where
S(A ) ) uA u ) ∑∑[A ]ij ) ∑(Vi) 2
2 T
2
i
j
S(AD) ) uADu ) ∑∑[AD]ij ) ∑(Vidi) i
j
(3) (4)
i
The term S(A2) is just the first Zagreb group index, while S(AD) is the true Schultz index (i.e., the nontrivial part of MTI), then reinvented by others.19,21 The parameter di (eq 4) stands for the ith row sum of entries in the distance matrix: di ) ∑i[D]ij. In the above relations, u and uT are the unit vector (of order N, which is the number of vertices in G) and its transpose, respectively, as recently used by Estrada et al.19,20 for rejecting the double sum symbol. In acyclic structures, there exists12,14,15 a linear correlation between the number S(AD) and the Wiener index
S(AD) ) 4W - N(N - 1)
i
(7)
j
W ) (1/2)∑∑[D]ij ) (1/2)uDuT i
(i,j)
(8)
j
S(AD) ) ∑[Ni∑Vk + Nj∑Vk] 2
i
T
W ) (1/2)∑∑[We]ij ) (1/2)uWeuT
k∈I
(9)
k∈J
where ∑k∈I and ∑k∈J denote the summation over all vertices lying on the i-side and j-side (i.e., to the I and J fragments, respectively) of the edge (i,j). Other valency-distance indices, composing two or three matrices, have been subsequently proposed.15,19 The present paper proposes a joint of the Cramer- and the Hadamard-matrix algebra by means of the W(M1,M2,M3) matrix. On this ground an extension of the definition of MTI and S(AD) indices is proposed. SCHULTZ-TYPE INDICES AND WALK MATRIX, W(M1,M2,M3)
A Schultz-type index, built up on a product of square matrices (of dimension N × N), one of them being obligatory the adjacency matrix, can be written (in Cramer matrix algebra) as
(5)
S in symbols of the type S(AD) reminds the name of Schultz. †
“Babes-Bolyai” University. E-mail address:
[email protected]. Drake University. X Abstract published in AdVance ACS Abstracts, October 1, 1997. ‡
S0095-2338(97)00019-X CCC: $14.00
MTI(M1,A,M3) ) uM1(A + M3)uT ) u(M1A + M1M3)uT ) S(M1A) + S(M1M3) (10) It is easily seen that MTI(A,A,D) is the Schultz original index. Analogue Schultz indices of sequence, (D,A,D), © 1997 American Chemical Society
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Chart 1. Schultz Index by Cramer Matrix Algebra for the Graph G1
(RD,A,RD), and (Wp,A,Wp), have been proposed and tested for correlating ability.16,17,20,25 In the above sequence, RD represents the matrix whose nondiagonal entries are 1/[D]ij; Wp is the Wiener-path matrix, i.e., the matrix defined by relation 6 when (i,j) represents a path. Cramer-matrix algebra concerning eqs 2 and 10 (M1 ) M3) is illustrated for the graph G1 (2,3-methylpentane) in the Charts 1 and 2.
AND
Chart 2. Matrices We, AD, and AWe for the Graph G1
sum in the matrix M1); [M2]ij gives the length of the walk, and the factor [M3]ij is taken from a third square matrix. Its diagonal entries are zero. It is a square, (in general) nonsymmetric matrix, of dimension N × N and was illustrated in refs 26 and 27. In the form [1/W(A,D,1)]ij (where 1 is the matrix having its (ij)-entries equal to 1 if i * j and zero otherwise), it has been defined in connection with the Harary-type indices28 and proved to be identical to the “restricted random walk matrix” proposed by Randic´.29 In fact, this matrix is a true operator, as will be shown below. Let, first, the combination (M1,M2,M3) be (M1,1,1). In this case, each column in the matrix W(M1,1,1) is identical to the vector RS(M1) (i.e., the product M1uT ) uM1T) except the (ii)-entries which are zero. Next, consider the combination (M1,1,M3); the corresponding walk matrix results as a Hadamard (pairwise) product30 (i.e., [Ma‚Mb]ij ) [Ma]ij[Mb]ij)
W(M1,1,M3) ) W(M1,1,1)‚M3 Walk matrix, W(M1,M2,M3), was defined,25-27 according to the principle of a single endpoint characterization of a path, as
[W(M1,M2,M3)]ij ) [M2]ijWM1,i[M3]ij
(11)
(12)
Recall that the sum of all entries in the Cramer product matrix, M1M3, can be achieved by multiplying their RS and CS vectors (i.e., the sum of entries on rows and columns, respectively)
∑∑[M1M3]ij ) CS(M1) RS(M3) ) uM1M3uT i
where WM1,i is the walk degree26 of the vertex i, weighted by the property collected in the matrix M1 (i.e., the ith row
RANDICÄ
(13)
j
Between W(M1,1,M3) and the Cramer product M1M3 there
W(M1,M2,M3)
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J. Chem. Inf. Comput. Sci., Vol. 37, No. 6, 1997 1097
exist the following relations
Chart 3. Schultz-Type Indices by W(M1,M2,M3) Algebra for the Graph G1
CS(W(M1T,1,M3)) ) uW(M1T,1,M3) ) CS(M1M3) ) uM1M3 (14) CS(W(M3,1,M1T)) ) uW(M3,1,M1T) ) RS(M1M3) ) M1M3uT (15) RS(W(M1T,1,M3)) ) W(M1T,1,M3)uT ) RS(M1T)‚RS(M3) (16) RS(W(M3,1,M1T)) ) W(M3,1,M1T)uT ) RS(M3)‚RS(M1T) (17) RS(M1T)‚RS(M3) ) RS(M3)‚RS(M1T)
(18)
From (16) and (13) one can write
uW(M1T,1,M3)uT ) u(RS(M1T)‚RS(M3)) ) uM1M3uT ) CS(M1)RS(M3) (19) Equations 16-18 show that the product within the matrix W(M1T,1,M3) is commutative. Equation 19 is the most important among the above six relations: it joins the Hadamard- and the Cramer-matrix algebra by means of the W(M1,M2,M3) matrix. Moreover, it suggests that the product matrix, M1M3, can also be viewed as a collection of pairwise products of local properties (encoded as row sums). These relations are illustrated, for the graph G1, in Chart 3. A condition for identical W(M1,1,M3) matrices (for one and the same graph) can be formulated.
W(M1,1,M3) ≡ W(M′1,1,M3) S RS(M1) ≡ RS(M′1) (20) Examples of identical W(M1,1,M3) matrices will be given below. A condition for degeneracy (i.e., a single value for two different graphs, Ga and Gb) of the sum of entries in the product matrix M1M3 can be drawn from eqs 18 and 19
RS(M1T(Ga))‚RS(M3(Ga)) ) RS(M3(Gb))‚RS(M1T(Gb)) (21) From relation 19 it is obvious that the same condition is also true for the degeneracy within the W(M1,1,M3) matrix. The walk matrix, W(M1,M2,M3), can be related to the Schultz numbers (cf. eq 10) as follows:
S(M1A) ) uW(M1T,1,A)uT
(22)
S(M1M3) ) uW(M1T,1,M3)uT
(23)
MTI(M1,A,M3) ) u(W(M1T,1,A) + W(M1T,1,M3))uT (24) It is easily seen that eqs 10 and 24 are equivalent. Equation 23 can be extended for raising at a power n + 1 a square matrix M ) M1 ) M3 by the aid of the walk matrix
u(Mn+1)uT ) uW(MT,n,M)uT ) uW(M,n,MT)uT
(25)
where n is the matrix having the (ij)-entries [n]ij for i * j and zero for i ) j.
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Table 1. Wiener W, Hyper-Cluj CJp, Hyper-Szeged, SZp, and Some S(M2) Indices in Octane Isomers grapha
W
CJp
SZp
S(A2)
S(D2)
S(CJu2)
S(SZu2)
C8 2MC7 3MC7 4MC7 3EC6 25M2C6 24M2C6 23M2C6 34M2C6 3E2MC5 22M2C6 33M2C6 234M3C5 3E3MC5 224M3C5 223M3C5 233M3C5 2233M4C4
84 79 76 75 72 74 71 70 68 67 71 67 65 64 66 63 62 58
210 185 170 165 150 161 147 143 134 129 149 131 122 118 127 115 111 97
340 320 307 294 272 308 288 282 268 242 280 250 258 220 254 242 234 232
13 14 14 14 14 15 15 15 15 15 16 16 16 16 17 17 17 19
1848 1628 1512 1476 1360 1420 1312 1280 1208 1172 1316 1176 1096 1072 1128 1032 1000 868
1596 1396 1284 1248 1136 1206 1102 1072 1004 968 1112 978 906 880 940 850 820 706
2620 2430 2380 2302 2142 2286 2195 2154 2074 1858 2089 1939 1922 1702 1868 1784 1730 1570
a
AND
RANDICÄ
Chart 4. Schultz Index by W(M1,M2,M3) Algebra for the Graph G1
M ) methyl; E ) ethyl.
Values S(M1M3), S(M1A), and the corresponding indices MTI(M1,A,M3) (M1 ) M3) for octanes are listed in Tables 1 and 2; matrices involved in the calculation of MTI(M1,A,M3) indices, for the graph G1, are illustrated in Charts 3 and 4. SZEGED AND CLUJ MATRICES IN SCHULTZ-TYPE INDICES
Two unsymmetric matrices, Mu, (M ) SZ, Szeged and CJ, Cluj) have been recently proposed by one of us.27,31-34 They are defined according to the principle25-27,31,32 of single endpoint characterization of a path, (i,j) (having its endpoints the vertices i and j)
[Mu]ij ) Ni,pk(i,j) ) |Vi,pk(i,j)|
(26)
Vi,pk(ij) ) {V|V ∈ V(G); [De]iV < [De]jV}
(27)
Vi,pk(i,j) ) {V|V ∈ V(G); DiV < DjV; ph(i,V) ∩ pk(i,j) ) {i}; pk(i,j)| ) min} (28) Ni,pk(i,j) ) |Vi,pk(i,j)| is the number of elements of the set Vi,pk(i,j), for CJu, the maximum value, max|Vi,pk(i,j)|, is taken over all paths pk(i,j). Diagonal entries are zero. The symbol Ni,pk(i,j) remembers the definition of the Wiener index (cf. eq 6).
Table 2. S(M1A) and MTI(M1,A,M3)a Indices in Octane Isomers graphb
S(AD)
S(AWe)
S(CJuA)
S(SZuA)
(D,A,Dq)
(CJu,A,CJu)
(SZu,A,SZu)
MTI
C8 2MC7 3MC7 4MC7 3EC6 25M2C6 24M2C6 23M2C6 34M2C6 3E2MC5 22M2C6 33M2C6 234M3C5 3E3MC5 224M3C5 223M3C5 233M3C5 2233M4C4
280 260 248 244 232 240 228 224 216 212 228 212 204 200 208 196 192 176
322 324 318 316 306 326 320 318 314 308 330 322 320 314 332 326 324 338
301 292 283 280 269 283 274 271 265 260 279 267 262 257 270 261 258 257
371 363 359 357 348 355 350 346 341 345 347 338 334 326 339 327 323 311
3976 3516 3272 3196 2952 3080 2852 2784 2632 2556 2860 2564 2396 2344 2464 2260 2192 1912
3493 3084 2851 2776 2541 2695 2478 2415 2273 2196 2503 2223 2074 2017 2150 1961 1898 1669
5611 5223 5119 4961 4632 4927 4740 4654 4489 4061 4525 4216 4178 3730 4075 3895 3783 3451
306 288 276 272 260 270 258 254 246 242 260 244 236 232 242 230 226 214
a Schultz-type indices, MTI(M ,A,M ), are written as the sequence (M ,A,M ); the original Schultz index is written as MTI. b M ) methyl; E 1 3 1 3 ) ethyl.
W(M1,M2,M3)
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J. Chem. Inf. Comput. Sci., Vol. 37, No. 6, 1997 1099
When defined on edges, Ie is an index (e.g., SZe, the classical Szeged index); when defined on paths, Ip is a hyperindex (e.g., SZp and CJp). Note that, in acyclic graphs, SZe ) CJe ) W; also note that SZe ) CJe in any connected graph. For other relations see ref 34. Values of these indices in octanes are listed in Table 1.
Chart 5. Matrices SZu and CJu for the Graph G1
Coming back to the Schultz-type indices (eq 10), we consider the case of CJu in the sequence (M1,A,M3) ) (CJu,A,CJu). Since the Cramer product is not commutative, and since CJu is an unsymmetric matrix, there exists a strong reason (see eq 13) to write a Schultz-type index as
MTI(CJu,A,CJu) ) (u(CJu)AuT + uA(CJu)uT)/2 + u(CJu)2uT ) (u(CJu)AuT + u(CJuT)AuT)/2 + u(CJu)2uT ) S(CJuA) + S(CJu2) (34) By virtue of relations 31 and 32, in acyclic structures, relation 34 can be written as
Vertices of the set Vi,pk(i,j) are selected as follows: (i) for SZu, they obey the original condition formulated by Gutman35 for the Szeged index (eq 27). Such vertices lie closer to the vertex i, with respect to the path (ij). (ii) for CJu, the set Vi,pk(i,j) consists of the vertices lying closer to the vertex i, and external with respect to the path pk(i,j) (condition ph(i,V) ∩ pk(i,j) ) {i}, eq 28). Since in cycle-containing structures, various shortest paths, pk(i,j), could supply various sets Vi,pk(i,j), by definition, the (ij)-entries in the Cluj matrices are taken as max |Vi,pk(i,j)| (see above). Both matrices, SZu and CJu, are defined in any connected graph, in contrast to the Wiener matrix, defined only in acyclic graphs. The externality condition (see above) strongly differentiates the two matrices (and their invariants). For the graph G1 they are illustrated in Chart 5. The two unsymmetric matrices allow the construction of the corresponding symmetric matrices, Mp (defined on paths) and Me (defined on edges), by
Mp ) Mu‚MuT
(29)
Me ) Mp‚A
(30)
In acyclic structures, the Cluj matrices, CJe and CJp, are identical to the Wiener matrices, We and Wp. In cyclic graphs, CJe superimposes on SZe, while CJp is different from SZp. In trees, the Cluj matrix obeys the relations31
RS(CJu) ) RS(We)
(31)
CS(CJu) ) CS(D)
(32)
Thus, CJu is a chimera between D and We (see below). Several indices can be calculated27,31-34 on these matrices, either as the half-sum of their entries (a relation of the type 7) or by
Ie/p ) ∑[Mu]ij[Mu]ji (ij)
MTI(CJu,A,CJu) ) (uDAuT + uAWeuT)/2 + uDWeuT ) (S(DA) + S(AWe))/2 + S(DWe)
(35)
Since A, D, and We are symmetric matrices, it is obvious that S(DA) ) S(AD) and S(AWe) ) S(WeA). Bearing in mind relations 11 and 24, MTI(CJu,A,CJu) can be written as
MTI(CJu,A,CJu) ) (uW(CJuT,1,A)uT + uW(CJu,1,A)uT)/2 + uW(CJuT,1, CJu)uT (36) which gives the exact result of eq 34 or 35. Relations 34 and 36 allow the calculation of this index in any connected graph. Thus, irrespective of being calculated, by Cramer or Hadamard algebra, the index can be written as in (37). The equivalence of the Schultz-type terms is immediate.
MTI(CJu,A,CJu) ) S(CJuA) + S(CJu2) ) (S(DA) + S(AWe))/2 + S(DWe) (37) Equation 37 offers a MTI number which encloses the information of three matrices, as suggested by relation 10. Values MTI(CJu,A,CJu) (denoted in refs 27 and 34 as CJuI) are given in Table 2. Returning to eq 20 and considering again relations 31 and 32, some examples of identical walk matrices are given
W(We,1,A) ≡ W(CJu,1,A); W(D,1,A) ≡ W(CJuT,1,A) W(We,1,CJu) ≡ W(CJu,1,CJu); W(D,1,CJuT) ≡ W(CJuT,1,CJuT)
(33) Szeged matrix, SZu, behaves similarly:
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MTI(SZu,A,SZu) ) (u(SZu)AuT + uA(SZu)uT)/2 + u(SZu)2uT ) (u(SZu)AuT + u(SZuT)AuT)/2 + u(SZu)2uT (38) MTI(SZu,A,SZu) ) (uW(SZuT,1,A)uT + uW(SZu,1,A)uT)/2 + uW(SZuT,1, SZu)uT (39) MTI(SZu,A,SZu) ) S(SZuA) + S(SZu2)
(40)
Values MTI(SZu,A,SZu) (denoted in ref 34 as SZuI) are given in Table 2. Within a set of acyclic isomers, a very interesting property comes out from the definition of SZu, which is presented as follows. Conjecture: the sum, over all vertices in a graph, of the products between the valency of a vertex i and the number of vertices closer to i (than to any other vertex j) is a constant
uA(SZu)uT ) u(RS(A)‚RS(SZu)) )
( (N3 ) + (N3 + 1 )) (41)
22
In other words, the sum on the product to the left of the Szeged matrix SZu with the adjacency matrix is a constant. In contrast, the product to the right, u(SZu)AuT ) u(RS(A)‚CS(SZu)) is variable within a set of acyclic isomers. Schultz-type indices show good correlation34 with some physicochemical properties of octanes, in two-variable regression: boiling points (MTI(D,A,D) and MTI ) 0.953), critical pressure (MTI(CJu,A,CJu) and MTI ) 0.988; MTI(SZu,A,SZu) and χ ) 0.967, χ being the connectivity index36), octane number (MTI(CJu,A,CJu) and MTI ) 0.987). Note that the Schultz original index MTI(A,A,D) was written above as simple MTI. CONCLUSIONS
Walk matrix, W(M1,M2,M3), is an interesting operator which works according to the Hadamard-matrix algebra. It offers various global invariants, a function of matrix combinations. Thus, it can be used as an alternative to the Cramer-matrix algebra to calculate Schultz-type indices. Other properties and applications of this operator will be presented in a further study.37 ACKNOWLEDGMENT
Many thanks are expressed to Professors Ivan Gutman, University of Kragujevac, Yugoslavia, and Bazil Parv, “Babes-Bolyai” University, Cluj, Romania, for helpful discussions. REFERENCES AND NOTES (1) Wiener, H. Structural Determination of Paraffin Boiling Points. J. Am. Chem. Soc. 1947, 69, 17-20. (2) Schultz, H. P. Topological Organic Chemistry. 1. Graph Theory and Topological Indices of Alkanes. J. Chem. Inf. Comput. Sci. 1989, 29, 227-228. (3) Schultz, H. P.; Schultz, E. B.; Schultz, T. P. Topological Organic Chemistry. 2. Graph Theory, Matrix Determinants and Eigenvalues, and Topological Indices of Alkanes. J. Chem. Inf. Comput. Sci. 1990, 30, 27-29. (4) Schultz, H. P.; Schultz, T. P. Topological Organic Chemistry. 3. Graph Theory, Binary and Decimal Adjacency Matrices, and Topological Indices of Alkanes. J. Chem. Inf. Comput. Sci. 1991, 31, 144-147. (5) Schultz, H. P.; Schultz, E. B.; Schultz, T. P. Topological Organic Chemistry. 4. Graph Theory, Matrix Permanents, and Topological Indices of Alkanes. J. Chem. Inf. Comput. Sci. 1992, 32, 69-72. (6) Schultz, H. P.; Schultz, T. P. Topological Organic Chemistry. 5. Graph Theory, Matrix Hafnians and Pfaffnians, and Topological Indices of Alkanes. J. Chem. Inf. Comput. Sci. 1992, 32, 364-366.
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(7) Schultz, H. P.; Schultz, T. P. Topological Organic Chemistry. 6. Theory and Topological Indices of Cycloalkanes. J. Chem. Inf. Comput. Sci. 1993, 33, 240-244. (8) Schultz, H. P.; Schultz, E. B.; Schultz, T. P. Topological Organic Chemistry. 7. Graph Theory and Molecular Topological Indices of Unsaturated and Aromatic Hydrocarbons. J. Chem. Inf. Comput. Sci. 1993, 33, 863-867. (9) Schultz, H. P.; Schultz, E. B.; Schultz, T. P. Topological Organic Chemistry. 8. Graph Theory and Topological Indices of Heteronuclear Systems. J. Chem. Inf. Comput. Sci. 1994, 34, 1151-1157. (10) Schultz, H. P.; Schultz, E. B.; Schultz, T. P. Topological Organic Chemistry. 9. Graph Theory and Molecular Topological Indices of Stereoisomeric Compounds. J. Chem. Inf. Comput. Sci. 1995, 35, 864870. (11) Schultz, H. P.; Schultz, E. B.; Schultz, T. P., Topological Organic Chemistry. 10. Graph Theory and Topological Indices of Conformational Isomers. J. Chem. Inf. Comput. Sci. 1996, 36, 996-1000. (12) Klein, D. J.; Mihalic´, Z.; Plavsˇic´, D.; Trinajstic´, N. Molecular Topological Index: A Relation with the Wiener Index. J. Chem. Inf. Comput. Sci. 1992, 32, 304-305. (13) Mihalic´, Z.; Nikolic´, S.; Trinajstic´, N. Comparative Study of Molecular Descriptors Derived from the Distance Matrix. J. Chem. Inf. Comput. Sci. 1992, 32, 28-37. (14) Plavs´ic´, D.; Nikoli, S.; Trinajsti, N.; Klein, D. J. Relation between the Wiener Index and the Schultz Index for Several Classes of Chemical Graphs. Croat. Chem. Acta 1993, 66, 345-353. (15) Gutman, I. Selected Properties of the Schultz Molecular Topological Index. J. Chem. Inf. Comput. Sci. 1994, 34, 1087-1089. (16) Diudea, M. V. Novel Schultz Analogue Indices. Commun. Math. Comput. Chem. (MATCH) 1995, 32, 85-103. (17) Diudea, M. V.; Pop, C. M. A Schultz-Type Index Based on the Wiener Matrix. Indian J. Chem. 1996, 35A, 257-261. (18) Klavzˇar, S.; Gutman, I. A Comparison of the Schultz Molecular Topological Index with the Wiener Index. J. Chem. Inf. Comput. Sci. 1996, 36, 1001-1003. (19) Estrada, E.; Rodriguez, L.; Gutie´rrez, A. Matrix Algebraic Manipulation of Molecular Graphs. 1. Distance and Vertex-Adjacency Matrices. Commun. Math. Comput. Chem. (MATCH) 1997, 35, 145-156. (20) Estrada, E.; Rodriguez, L. Matrix Algebraic Manipulation of Molecular Graphs. 2. Harary- and MTI-Like Molecular Descriptors. Commun. Math. Comput. Chem. (MATCH) 1997, 35, 157-167. (21) Dobrynin, A. A.; Kochetova, A. A. Degree Distance of a Graph: A Degree Analogue of the Wiener Index. J. Chem. Inf. Comput. Sci. 1996, 36, 1082-1086. (22) Randic´, M.; Guo, X.; Oxley, T; Krishnapriyan, H. Wiener Matrix: Source of Novel Graph Invariants. J. Chem. Inf. Comput. Sci. 1993, 33, 700-716. (23) Randic´, M.; Guo, X.; Oxley, T.; Krishnapriyan, H.; Nayor, L. Wiener Matrix Invariants. J. Chem. Inf. Comput. Sci. 1994, 34, 361-367. (24) Hosoya, H. Topological Index. A Newly Proposed Quantity Characterizing the Topological Nature of Structural Isomers of Saturated Hydrocarbons. Bull. Chem. Soc. Jpn. 1971, 44, 2332-2339. (25) Diudea, M. V.; Ivanciuc, O. Molecular Topology (in Romanian); Comprex: Cluj, 1995. (26) Diudea, M. V. Walk Numbers eWM: Wiener-Type Numbers of Higher Rank. J. Chem. Inf. Comput. Sci. 1996, 36, 535-540. (27) Diudea, M. V. Cluj Matrix, CJu: Source of Various Graph Descriptors. Commun. Math. Comput. Chem. (MATCH) 1997, 35, 169-183. (28) Diudea, M. V. Indices of Reciprocal Property or Harary Indices. J. Chem. Inf. Comput. Sci., in press. (29) Randic´, M. Restricted Random Walk on a Graph as a Source of Novel Molecular Descriptors. Theor. Chem. Acta, in press. (30) Horn, R. A.; Johnson, C. R. Matrix Analysis; Cambridge University Press: Cambridge, U.K., 1985. (31) Diudea, M. V. Wiener and Hyper-Wiener Numbers in a Single Matrix. J. Chem. Inf. Comput. Sci. 1996, 36, 833-836. (32) Diudea, M. V. Cluj Matrix Invariants. J. Chem. Inf. Comput. Sci. 1997, 37, 300-305. (33) Diudea, M. V.; Minailiuc, O. M.; Katona, G.; Gutman, I. Szeged Matrices and Related Numbers. Commun. Math. Comput. Chem. (MATCH), 1997, 35, 129-143. (34) Diudea, M. V.; Parv, B.; Topan, M. I. Derived Szeged and Cluj Indices. J. Serb. Chem. Soc. 1997, 62, 267-276. (35) (a) Gutman, I. A Formula for the Wiener Number of Trees and Its Extension to Graphs Containing Cycles. Graph Theory Notes New York 1994, 27, 9-15. (b) During the preparation of this manuscript, Professor Ivan Gutman has proposed some notations in the view of simplifying and clarifying mathematical relations (see also: Gutman, I.; Diudea, M. V. Defining Cluj Matrices and Cluj Matrix Invariants. Commun. Math. Comput. Chem. (MATCH), submitted for publication. (36) Randic´, M. On Characterization of Molecular Branching. J. Am. Chem. Soc. 1975, 97, 6609-6615. (37) Diudea, M. V. Valencies of Property. Commun. Math. Comput. Chem. (MATCH), submitted for publication.
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