Scientific paper
Shell-Polynomials and Cluj-Tehran Index in Tori T(4,4)S[5,n]
Mircea V. Diudea1 and Ali Reza Ashrafi2
1 Faculty of Chemistry and Chemical Engineering, Babes-Bolyai University, 3400 Cluj, Romania 2 Department of Mathematics, Faculaty of Science, University of Kashan, Kashan 87317-51167,1. R. Iran * Corresponding author: E-mail: diudea@gmail.com Received: 28-04-2010 This paper is dedicated to Professor Milan Randi} on the occasion of his 80th birthday
Abstract
Weighted Hosoya polynomials have been developed by Diudea, in ref. Studia Univ. "Babes-Bolyai", 2002, 47, 131-139. Among various weighting schemes, those polynomials obtained by using Diudea's Shell matrix operator are far more interesting. We present here the Shell-Distance and Shell-Degree-Distance polynomials and close formulas to calculate them and derived Cluj-Tehran CT index in the family of square tiled tori T(4,4)S[5,n]. Applications of the proposed descriptors are also presented.
Keywords: Weighted Hosoya polynomial; Shell-polynomial; Cluj-Tehran index
1. Introduction
Let G(V,E) be a connected molecular graph,1 without directed and multiple edges and without loops, the vertex and edge-sets of which being represented by V(G) and E(G), respectively. Define the kth layer/shell of vertices v lying at distance k with respect to the reference vertex i as:2
The collection of all its layers defines the partition of G with respect to i:
with ecct being the eccentricity of i (i.e., the largest distance from i to the other vertices in G).
On the above notions, we define the entries in a layer matrix (of a vertex property) LM as:2-5
The zero column is just the column of vertex properties [LM]. 0 = p. Any atomic/vertex property can be considered as pi.
The information can be taken from any square matrix M (called here the info matrix) as row sum RS, column sum CS or diagonal entries given by the Walk matrix,2 as developed by TOPOCLUJ software package.6
The Layer matrix LM is a collection of the above defined entries:
LM = {[LM]/ lt; i e V(G); k e [0,1,.., d(G)]} (4)
with d(G) being the diameter of the graph or the largest distance in G.
Similarly, we can define the entries in a shell matrix ShM:24
where M is any square topological matrix. Any operation, (other than summation) over the square matrix entries [M]i,v, can be used, thus Sh being a true matrix operator. The shell matrix will collect the above defined entries:
ShM = {[ShM];>; i e V(G); k e [0,l,..,rf(G)]} (6)
The zero column [ShM]i 0 collects the diagonal entries in the info matrix M.
2. LM Polynomials
Define a distance-based polynomial as:7 P(x) = Ztp(G,k)-xl
(7)
with p(G,k) being sets of local contributions (of extent k) to the global (molecular) property P(G) = u p(G,k) and summation running up to d(G). The polynomial coefficients are calculable from the above defined layer/shell matrices, as the half sums on columns.
Some single number descriptors (i.e., topological indices TIs) can be calculated by evaluating the polynomial k derivatives (usually in x = 1):
P*(1) = £ .k\p(Gtk)
(8)
In case: p(v) = 1 (i.e., the vertex counting), LM = LC, (i.e., layer matrix of counting), p(G,k) denotes the number of pair vertices separated by distance k in G, and the classical Hosoya H(x) polynomial is recovered.8 The index calculated as the polynomial first derivative is the well-known Wiener index,9 W.
W(G) = P'(LC,l)
(9)
The hyper-Wiener index WW, patterned by Ran-dic,10 is calculated from P'(1) and P"(1) derivatives as:
ffW(G) = P'(LC,l) + (l/2)/>*(LC,l)	(10)
Figure 1 illustrates the P(LC,x) polynomial.
i/k	1	2	3	4
\	1	2	2	!
2	3	2	!	0
3	3	3	0	0
4	?	2	2	0
5	1	1	2	2
6	1	2	2	1
7	1	2	3	0
CS	12	14	12	4
Figure 1. The graph G1 and its Hosoya polynomial P(LC,x) = 6x + 7x2 + 6x3 + 2x4; P'(LC,1) = W = 46
tion is focused on Shell-Distance and Shell-Degree-Distance polynomials, respectively.
3. 1. Shell-Distance Polynomial
Define the Shell-Distance polynomial7,11 as
where p(G,k) are calculated as the column half sums in the matrix ShD (Table 1). The polynomial defined on ShD matrix has the coefficients already multiplied by the (to-pological) distance and clearly P(ShD,1) = W(G), the well-known Wiener index.
The 1st derivative of this polynomial is then weighted by the squared distances in G. An index, called Cluj-
Table 1. Polynomial P(ShD,x) and corresponding CT index in G1.
P(1)
P(1) P"(1)
CT (ShD)
ShD
D
i \ k	1	2	3	4	RS		1	2	3	4	5	6	7	RS
1	1	4	6	4	16		0	1	2	3	4	2	3	15
2	3	4	3	0	11		1	0	1	2	3	1	2	10
3	3	6	0	0	10		2	1	0	1	2	2	1	9
4	2	4	6	0	13		3	2	1	0	1	3	2	12
5	1	2	6	8	18		4	3	2	1	0	4	3	17
6	1	4	6	4	16		2	1	2	3	4	0	3	15
7	1	4	9	0	15		3	2	1	2	3	3	0	14
CS P(x)	12 6x	28 14x2	36 18x3	16 8x4	92	CS	15	10	9	12	17	15	14	92
46 120 232 236
W
3. ShM Polynomials
The weighting in the Shell polynomials is performed by the aid of Diudea's Shell matrix operator2,4 and the weight is just the property of info matrix M. Our atten-
Tehran11 CT(ShM) index (with specified M) is calculated by relation
CT{ShM. G) = P'(ShM, !) + (!/ 2)/5*(ShM, 1 ) (12)
The results, for G1, are given at the bottom of Table 1.
3. 2. Shell-Degree-Distance Polynomials
The Cramer product of the diagonal matrix of vertex degrees Deg with the Distance D matrix provides the matrix of degree distances, denoted DegD.
Table 2. Polynomial P(ShDegD,x) and corresponding CT index in G1.
The above Cramer product is equivalent (gives the same half sum of entries) with the pair-wise (Hadamard) product of the vectors "row sum" RS in the Adjacency A and Distance D matrices, respectively12
RS(A) *RS(D) = /?S(DegD)	(14)
Next, applying the Shell operator, the matrix Sh-DegD, of which column half sums are just the coefficients of the corresponding Shell-polynomial, is obtained
ShDegD	DegD
i \ k	1	2	3	4	RS		1	2	3	4	5	6	7	RS
1	1	4	6	4	15		0	1	2	3	4	2	3	15
2	9	12	9	0	30		3	0	3	6	9	3	6	30
3	9	18	0	0	27		6	3	0	3	6	6	3	27
4	4	8	12	0	24		6	4	2	0	2	6	4	24
5	1	2	6	8	17		4	3	2	1	0	4	3	17
6	1	4	6	4	15		2	1	2	3	4	0	3	15
7	2	4	9	0	14		3	2	1	2	3	3	0	14
CS	26	52	48	16	142	CS	24	14	12	18	28	24	22	142
P(x)	13x	26x2	24x3	8x4										
P(1)	71
P'(1)	169
P"(1)	292
CT (ShDegD)	315
Table 3. Matrix operations of the diagonal Degree Deg matrix
Deg x D = DegD	D x Deg = DDeg
	1	2	3	4	5	6	7	RS		1	2	3	4	5	6	7	RS
1	0	1	2	3	4	2	3	15	1	0	3	6	6	4	2	3	24
2	3	0	3	6	9	3	6	30	2	1	0	3	4	3	1	2	14
3	6	3	0	3	6	6	3	27	3	2	3	0	2	2	2	1	12
4	6	4	2	0	2	6	4	24	4	3	6	3	0	1	3	2	18
5	4	3	2	1	0	4	3	17	5	4	9	6	2	0	4	3	28
6	2	1	2	3	4	0	3	15	6	2	3	6	6	4	0	3	24
7	3	2	1	2	3	3	0	14	7	3	6	3	4	3	3	0	22
CS	24	4	12	18	28	24	22	142	CS	15	30	27	24	17	15	14	42
1/2SUM =		71			1/2SUM		= 71										
Sh(DegD)									Sh(DDeg)								
	0	1	2	3	4	RS				0	1	2	3	4	RS		
1	0	1	4	6	4	15			1	0	3	8	9	4	24		
2	0	9	12	9	0	30			2	0	5	6	3	0	14		
3	0	9	18	0	0	27			3	0	6	6	0	0	14		
4	0	4	8	12	0	24			4	0	4	8	6	0	18		
5	0	1	2	6	8	17			5	0	2	6	12	8	28		
6	0	1	4	6	4	15			6	0	3	8	9	4	24		
7	0	1	4	9	0	14			7	0	3	10	9	0	24		
CS	0	26	52	48	16	1 42			CS		26	52	48	16	142		
P(1)		13	26	24	8	71			P(1)		13	26	24	8	71		
P'(1)		13	52	72	32	169			P'(1)		13	52	72	32	169		
An example is given in Table 2; at the bottom of table, CT index is given.
Irrespective CP (11) is performed "to the left" or "to the right", the Shell-polynomial is the same (Table 3).
The half sum of entries in the Deg x D or D x Deg matrices is the well-known degree-distance DegD(G) index, defined by Dobrynin and Kochetova13
where Deg(v) and D(v) are just RS(A(v)) and RS(D(v)), see (13). This index is in fact the non-trivial part of the Schultz index.14-26 Accordingly, this index can be calculated as the half sum of entries in the matrices A x D or D x A (Table 4). Next, applying the Shell operator, we obtain the matrices ShA x D/ShD x A which are different from ShDeg x D / ShD x Deg, because of the non-zero diagonal of A x D/D x A, of which information is lost in the further first derivative of the corresponding Shell-polynomial. Even the P(1) values are the same and equal the degree-distance index DegD(G), in the following we will calculate only the polynomial P(ShDegD,x).
Another reason is that the entries in DegD matrix represent precisely the property defined by Dobrynin in
(14). This matrix can also be obtained by Diudea's Walk operator12,16 as
where 1 stands for the square matrix of the pertinent order, having all the non-diagonal entries 1 while the diagonal entries zero.
The walk operator4,5,12,16 W(M1,M2,M3) is defined as [W(M,, M2, M3 )], ^ = RS(M/M"lu ),- [M3 \ j (19)
It works by Hadamard algebra and was extensively exemplified in refs.12,16 (see also ref.27)
4. Shell Polynomials P(ShD,x) and P(ShDegDx) in T(4,4)S[5,w]
The above defined Shell-polynomials are calculated for the series of square tiled nanotori T(4,4)S[5,n] (Figure 2). Because such graphs are vertex transitive, formulas are given only for the vertex polynomial. At the end of each case, examples are given for the Cluj-Tehran CT index. To obtain the global value of CT the value per vertex is to be multiplied by half the number of vertices in the torus: v = 5n.
Table 4. Matrix operations involving AxD matrix
A x D	D x A
	1	2	3	4	5	6	7			1	2	3	4	5	6	7	
1	1	0	1	2	3	1	2	10	1	1	4	7	6	3	1	2	24
2	4	3	4	7	10	4	7	39	2	0	3	4	4	2	0	1	14
3	7	4	3	4	7	7	4	36	3	1	4	3	2	1	1	0	12
4	6	4	2	2	2	6	4	26	4	2	7	4	2	0	2	1	18
5	3	2	1	0	1	3	2	12	5	3	10	7	2	1	3	2	28
6	1	0	1	2	3	1	2	10	6	1	4	7	6	3	1	2	24
7	2	1	0	1	2	2	1	9	7	2	7	4	4	2	2	1	22
24 14 12 18 28 24 22 142	10 39 36 26 12 10 9 142
1/2SUM = 71	1/2SUM = 71
Sh(A x D )	Sh(D x A)
0 1 2 3 4 RS	0 1 2 3 4 RS
1	1	0	2	4	3	10
2	3	12	14	10	0	39
3	3	12	21	0	0	36
4	2	4	8	12	0	26
5	1	0	1	4	6	12
6	1	0	2	4	3	10
7	1	0	2	6	0	9
CS 12 28 50 40 12 142
P(1) 614 25 20 6 71 P'(1) 14 50 60 24 148
1	1	4	8	8	3	24
2	3	4	5	2	0	14
3	3	6	3	0	0	12
4	2	4	8	4	0	18
5	1	2	7	12	6	28
6	1	4	8	8	3	24
7	1	4	11	6	0	22
CS 12 28 50 40 12 142
P(1) 6 14 25 20 6 71 P'(1) 14 50 60 24 148
Figure 2. Torus of the series T(4,4)S[5,n],
4. 1. Case: n = 5 + 10 m.
,5(m-I>+9
+ (24 + 40m) ■	+{16 + 2pm) x
i 2125m + 925m +-+172
12625m1 8775m" 6245m L« h---1--+-+ 368
Example:
C7,(ShDI(/J7,(4,4)S[5,5+10m])) :
m(l;2;3) = 8526; 489%; 165016
4. 2. Case: n = 10 m.
JP(ShDI(/,r(4,4)5[5,10m]) = 4A- + 16^+30.v3 + 40^ + ¿¿[50(^-t) + 10i-]-A-5,i-|)+r+[45 + 45(m-D|-Jc5<"'-!,+5 +
[36 + 30(m- l)]-.v:
J(m-]>+6
+ [14 + 10(m-l)]x
Sim-] 1+7
Cr(ShDI(js r(4, 4)jS[5, 10m])) =
5m(1875m3 + 2300m2 + 1335m + 382) 12
Example: Cr(ShDl(i,T(4A)S[5,l0m])):
m(2,3,4) = 22710; 94640; 270870
Because in T(4,4)S[5,n] the vertices have all the degree 4, all the above formulas must be multiplied by 4 to get the corresponding degree-distance descriptors.
5. Modeling Octane Properties
To test the correlating ability of the descriptors derived from the degree-distance matrices (ShMt) and Shell-polynomials we focused on the set of octanes, as one of the benchmark-sets28-30 in correlating studies by using to-pological indices.
Figure 3. The plot of octanes BP (°C) vs DegD index
Among several physico-chemical properties, the boiling point BP was best modeled, in monovariate regression, by the index DegD (plotted in Figure 3) while in trivariate model, by the other descriptors derived from the Shell-polynomials, the Cluj-Tehran index included. Remark the best monovariate model reported so far is only 0.78 for the BP of octanes.28 Other applications refer to the discriminating studies in large databases comprising molecular graphs: the super index CJN (Cluj-Nis), including descriptors derived from the Shell-polynomials, succeeded in solving a set of more than five thousands molecules without degeneracy.11
6. Conclusions
Weighting a distance-based polynomial is easily achieved by using the Shell-matrix operator. This enables one to calculate a variety of so-called Shell-polynomials. In this article we derived close formulas to calculate the Shell-polynomial P(ShD,x) and P(ShDegD,x) in a family of square tiled tori. Applications of the proposed descriptors demonstrated their utility.
7. Acknowledgements
The work was supported by the Romanian Grant CNCSIS PN-IIIDEI 506/2007
8. References
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Povzetek
Utežene Hosoya polinome je razvil Diudea (v ref.: Studia Univ. "Babes-Bolyai", 2002, 47, 131-139). Med raznolikimi utežitvenimi shemami so precej bolj zanimivi polinomi, dobljeni z uporabo Diudejevega operatorja matrike lupin (obodov) grafa. Tu so predstavljeni polinomi obodnih razdalj in stopnje obodnih razdaj (Shell-Distance, Shell-Degree-Distance) in izpeljane formule za njihov izračun ter za Cluj-Tehran CT indeks v družini »kvadratno-razdeljenih« torusov T(4,4)S[5,n]. Predstavljena je tudi uporaba predlaganih deskriptorjev.