The most used molecular graph descriptors in establishing Quantitative structure-property relationships (QSPRS) and Quantitative structure-activity relationships (QSARS) are topological indices. Molecular descriptors are normally chosen based on their ability to give good results in statistical models. In this paper we introduce a set of five new indices (Kekule indices) K, K1, K2, K3, K4 and we establish that the Kekule index (K) has excellent correlation (r = 0.99999997250969) with log p values in case of polyacenes

Lakshmi, K Lavanya

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Journal of Experimental Sciences 2012, 3(4): 18-21 ISSN: 2218-1768 Available Online: http://jexpsciences.com/    A highly correlated topological index for polyacenes  K. Lavanya Lakshmi  Department of mathematics, Bapatla Engineering College, Bapatla, Guntur (D.t), Andhra Pradesh, 522101, India  Abstract  The most used molecular graph descriptors in establishing Quantitative structure-property relationships (QSPRS) and Quantitative structure-activity relationships (QSARS) are topological indices. Molecular descriptors are normally chosen based on their ability to give good results in statistical models. In this paper we introduce a set of five new indices (Kekule indices) K, K1, K2, K3, K4 and we establish that the Kekule index (K) has excellent correlation (r = 0.99999997250969) with log p values in case of polyacenes.  Keywords: Kekule index, Polyacenes.  INTRODUCTION       A topological index of a chemical compound is an integer, derived following a certain rule, which can be used to characterize the chemical compound. The first topological index is Wiener index introduced by Harold Wiener in 1947 to demonstrate its relation with physicochemical properties of alkanes, alcohols and amines. Ever since it is known that topological indices can be used to establish Quantitative structure - property relationships (QSPRS) and Quantitative structure - activity relationships (QSARS) in pharmacology, researchers are pursuing several investigations to find topological indices having correlation one or nearer to one with physicochemical properties of organic compounds. In this paper we introduce a new index K (Kekule index) having correlation (r = 0.99999997250969) with log p values of polyacenes.                                                      DEFINITIONS   In this section we define five new topological indices and explain the procedure of computation.  Kekule index: The Kekule index of a graph G = (V, E) is defined as  K (G) =( )( )e uv E GW e= ∈∑ where W (e) = |i - j|, i, j are the degrees of the vertices u and v in G.  K1 index: The K1 index of a graph G = (V, E) is defined as K1 (G) =( )( )e uv E GW e= ∈∑   where W (e) = |i - j|2, i, j are the degrees of the vertices u and v in G.  K2 index: The K2 index of a graph G = (V, E) is defined as K2 (G)  =( )( )e uv E GW e= ∈∑ where W (v) = sum of the degrees of the neighboring vertices in G.  K3 index: The K3 index of a graph G = (V, E) is defined as K3 (G)   =( )( )e uv E GW e= ∈∑ where W(v)=product of the degrees of the neighboring vertices in G.  K4 index: The K4 index of a graph G = (V, E) is defined as K4 (G)   =( )( )e uv E GW e= ∈∑ where W (v) = max { }( , ) / ( )i id v v v V G∈ . Where    d (v, vi) is the maximum distance between v and vi.       We compute these indices considering the chemical graph of the compound Naphthalene given below.  Fig 1. Graph of Naphthalene  Calculation of K index: W(e1)=|2-2|=0, W(e2)=|2-3|=1, W(e3)=|3-2|=1, W(e4)=|2-2|=0, W(e5)=|2-2|=0, W(e6)=|2-2|=0,   W(e7)=|3-2|=1, W(e8)=|3-2|=1, W(e9)=|2-2|=0, W(e10)=|2-2|=0,  W (e11)=|3-3|=0 Therefore K(G)=1+1+1+1=4. Calculation of K1 index: W(e1)=|2-2|2=0, W(e2)=|2-3|2=1, W(e3)=|3-2|2=1, W(e4)=|2-2|2=0, W(e5)=|2-2|2=0,  W(e6)=|2-2|2=0, W(e7)=|3-2|2=1, W(e8)=|3-2|2=1, W(e9)=|2-2|2=0, W(e10)=|2-2|2=0,   W (e11) = |3-3|2=0 Therefore K1 (G) = 1 1 1 1+ + + =2.   Received:; Revised; Accepted.  *Corresponding Author K Lavanya Lakshmi  Department of mathematics, Bapatla Engineering College, Bapatla, Guntur (D.t), Andhra Pradesh, 522101, India   Email: k.lavanya@yahoo.co.in    Journal of Experimental Sciences 2012, 3(4): 18-21  19Calculation of K2 index: W(v1)= 2+2=4, W(v2)= 2+3=5, W(v3)= 2+3+2=7, W(v4)= 3+2=5, W(v5)= 2+2=4, W(v6)= 2+2=4, W(v7)= 2+3=5, W(v8)= 2+2+3=7, W(v9)= 3+2=5, W(v10)= 2+2=4,   Therefore K2 (G) = 4+5+7+5+4+4+5+7+5+4=50. Calculation of K3 index: W(v1)= 2 × 2=4, W(v2)= 2 × 3=6, W(v3)= 2 × 3 × 2=12, W(v4)= 3× 2=6, W(v5)= 2× 2=4, W(v6)= 2 × 2=4, W(v7)= 2 × 3=6, W(v8)= 2 × 2 × 3=12, W(v9)= 3× 2=6, W(v10)= 2× 2=4,   Therefore K3 (G) = 4+6+12+6+4+4+6+12+6+4=64. Calculation of K4 index: W (v1) = 5, W (v2) = 4, W (v3) = 3, W (v4) = 4, W (v5) = 5, W (v6) = 5, W (v7) = 4,  W (v8) = 3, W (v9) = 4, W (v10) = 5,   Therefore K4(G)=5+4+3+4+5+5+4+3+4+5=42.  NOTATION       The molecular graph of polyacenes is a chain of hexagons arranged linearly. For convenience we adopt the following notation: Let L (a) be the graph consisting of a hexagons in one row as shown in the figure below. Here a is a positive integer.  Fig 2. Graph of polyacene with a hexagons in 1 row MAIN RESULTS   Throughout the paper we take   { } { }{ } { } { }1 2 2 1 1 2 2 11 1 2 1 3, ,..., , , ,..., ./ 1, 2,..., 2 , / 1, 2,..., 2 , / 1,3,..., 2 1a ai i i i i iA x x x B y y yE x x i a E y y i a E x y i a+ ++ += == = = = = = +  Theorem 4.1: If L(a) is the chemical graph (see Figure 2) then the K index of the  graph L(a) is K(L(a)) =4(a-1), where a is a positive integer.  Proof: Consider  { } { } { }1 1 1 2 3( ( )) ( ) ( ) ( )i i i i i ie x x E e y y E e x y EK L a W e W e W e+ += ∈ = ∈ = ∈= + +∑ ∑ ∑   Here  { }1 1( ) (2 2)1i ie x x EW e a+= ∈= −∑    (since there are 2a-2 edges with | i-j|=1 in E1).  { }1 2( ) (2 2)1i ie y y EW e a+= ∈= −∑    (since there are 2a-2 edges with | i-j|=1 in E2).   { } 3( ) 0i ie x y EW e= ∈=∑  (since for every edge in E3,  | i-j|=0). Therefore ( ( )) 2 2 2 2 4 4 4( 1)K L a a a a a= − + − = − = − .  Theorem 4.2: If L(a) is the chemical graph (see Figure 2) then the K1 index of the graph L(a) is K1(L(a)) =2 ( 1)a − , where a is a positive integer.  Proof: Consider { } { } { }1 1 1 2 31( ( )) ( ) ( ) ( )i i i i i ie x x E e y y E e x y EK L a W e W e W e+ += ∈ = ∈ = ∈= + +∑ ∑ ∑   here  { }1 1( ) (2 2)1i ie x x EW e a+= ∈= −∑    (since there are 2a-2 edges with | i-j|2=1 in E1).   { }1 2( ) (2 2)1i ie y y EW e a+= ∈= −∑    (since there are 2a-2 edges with | i-j|2=1 in E2).   { } 3( ) 0i ie x y EW e= ∈=∑   (since for every edge in E3,  | i-j|2=0).  Therefore 1( ( )) 2 2 2 2 4 4 2 ( 1)K L a a a a a= − + − = − = −   Theorem 4.3: If L(a) is the chemical graph (see Figure 2) then the K2 index of the graph L(a) is K2(L(a)) =26a-2, where a is a positive integer.  Proof: In the set A two vertices namely 1x  and 2 1ax +  having weight 4, two vertices namely 2x  and 2ax  having weight 5, (a-2) vertices namely 4 6 8 2 2, , ,..., ax x x x −  having weight 6, (a-1) vertices namely 3 5 7 2 1, , ,..., ax x x x −  having weight 7.      In the set B two vertices namely 1y  and 2 1ay +  having weight 4, two vertices namely 2y  and 2ay  having weight 5, (a-2) vertices namely 4 6 8 2 2, , ,..., ay y y y −  having weight 6, (a-1) vertices namely 3 5 7 2 1, , ,..., ay y y y −  having weight 7. Therefore K2(L(a)) = { } { }( ) ( )i ii ix A y BW x W y∈ ∈+∑ ∑  Here   { }{ }( ) 2(4) 2(5) ( 2)6 ( 1)7( ) 2(4) 2(5) ( 2)6 ( 1)7iiix Aiy BW x a aW y a a∈∈= + + − + −= + + − + −∑∑      Therefore     2 ( ( )) 2(4) 2(5) ( 2)6 ( 1)7 2(4) 2(5) ( 2)6 ( 1)7 26 2K L a a a a a a= + + − + − + + + − + − = − .  Theorem 4.4: If L(a) is the chemical graph (see Figure 2) then the K3 index of the graph L(a) is K3(L(a)) = 24 when a = 1= 42a-20, where a>1 is a positive integer. K.Lavanya Lakshmi   20Proof: In the set A two vertices namely 1x  and 2 1ax +  having weight 4, two vertices  namely 2x  and 2ax  having weight 6, (a-2) vertices namely 4 6 8 2 2, , ,..., ax x x x −  having weight 9, (a-1) vertices namely 3 5 7 2 1, , ,..., ax x x x −  having weight 12.      In the set B two vertices namely 1y  and 2 1ay +  having weight 4, two vertices namely 2y  and 2ay  having weight 6, (a-2) vertices namely 4 6 8 2 2, , ,..., ay y y y −  having weight 9, (a-1) vertices namely 3 5 7 2 1, , ,..., ay y y y −  having weight 12. Therefore K3(L(a)) = { } { }( ) ( )i ii ix A y BW x W y∈ ∈+∑ ∑  Here { }{ }( ) 2(4) 2(6) ( 2)9 ( 1)12( ) 2(4) 2(6) ( 2)9 ( 1)12iiix Aiy BW x a aW y a a∈∈= + + − + −= + + − + −∑∑  Therefore  3 ( ( )) 2(4) 2(6) ( 2)9 ( 1)12 2(4) 2(6) ( 2)9 ( 1)12 42 20K L a a a a a a= + + − + − + + + − + − = − .  Theorem 4.5: If L(a) is the chemical graph (see Figure 2) then the K4 index of the graph L(a) is K4(L(a)) = 18 when a = 1= 6a2+8a+2, where a>1 is a positive integer.  Proof:  We know K4 (G) =( )( )v V GW v∈∑  K4(L(a)) = { } { }( ) ( )i ii ix A y BW x W y∈ ∈+∑ ∑ Here    { }[ ]{ }[ ]( ) 2 (2 1) (2 ) (2 1) ... (2 ( 2)) (2 ( 1))( ) 2 (2 1) (2 ) (2 1) ... (2 ( 2)) (2 ( 1))iiix Aiy BW x a a a a a a aW y a a a a a a a∈∈= + + + − + + − − + − −= + + + − + + − − + − −∑∑   Therefore   [ ][ ]4 ( ( )) 2 (2 1) (2 ) (2 1) ... (2 ( 2)) (2 ( 1))2 (2 1) (2 ) (2 1) ... (2 ( 2)) (2 ( 1))K L a a a a a a a aa a a a a a a= + + + − + + − − + − − ++ + + − + + − − + − −             = [ ]2 2( )(2 ) 2(2 3 ... ( 2)) 2( 1)a a a a− + + + − + +              = 22 4 ( 2)( 1) 2 (2 2)a a a a − − − + + +               = 26 8 2a a+ + .   Remark 4.6: K1, K2, K3 can be represented in terms of K as follows:              1 2 326 42, 24, 224 4K K K K K K= = + = +  (where 1a ≠  in K3)  COMPARISON OF RESULTS        In this section we compute the correlation of log p with K, K1, K3 and K4 considering the first 20 compounds of polyacenes (L(a), a=1 to 20)      The values of K, K1, K2, K3, K4 and log p for these compounds are tabulated below (Table 1).   Table 1.  No K K1 K2 K3 K4 log p 1 0 0 24 24 18 2.202 2 4 2 50 64 42 3.396 3 8 2.828 76 106 80 4.590 4 12 3.464 102 148 13 5.784 5 16 4 128 190 192 6.978 6 20 4.472 154 232 266 8.172 7 24 4.8989 180 274 352 9.366 8 28 5.2915 206 316 450 10.560 9 32 5.657 232 358 560 11.754 10 36 6 258 400 682 12.948 11 40 6.3245 284 442 816 14.142 12 44 6.6332 310 484 962 15.336 13 48 6.928 336 526 1120 16.530 14 52 7.211 362 568 1290 17.724 15 56 7.4833 388 610 1472 18.918 16 60 7.7459 414 652 1666 20.112 17 64 8 440 694 1872 21.306 18 68 8.2642 466 736 2090 22.500 19 72 8.4852 492 778 2320 23.694 20 76 8.7178 518 820 2562 24.880 Following are correlation coefficients of log p with K, K1, K3 and K4 Table 2.  r log p K 0.99999997250969 K1 0.96545941249478 K3 0.99999854115530 K4 0.97445287313308                                                 CONCLUSIONS         From the table it is clear that K is highly correlated with log p compared to other indices. In [10] we observed that among W, PI, Sz, Sh and Fr indices of polyacenes, Sh7 is highly correlated with log p with the value of r = 0.9996. However our new index K has better correlation 0.99999997250969 (almost one) than Sh index and this is more suitable for QSAR/QSPR studies. The regression equation between log p and K is   log p = 2.20268571428571 + 0.29847142857143 K and the graphs of log p and predicted log p are given below. 0 2 4 6 8 10 12 14 16 18 2002468101214161820nPredLogP Fig 3 (a). Graph of predicted log p   Journal of Experimental Sciences 2012, 3(4): 18-21  210 2 4 6 8 10 12 14 16 18 2002468101214161820nLogP Fig 3 (b). Graph of log p ACKNOWLEDGMENT       I thank my research supervisor Prof. N. Prabhakara Rao for his encouragement and guidance in the preparation of this paper.  REFERENCES  [1] Ali Reza Ashrafi, Amir Loghman, 2006. PI index of some Benzenoid graphs, Journal of Chilean Chemical Society, 51(3)(2006), 968-970. [2] Damir Verkicevic, Nenod Trinajstic, 2004. Wiener index of Benzenoid graphs, Bulletin of Chemists and Technologists of Macedonia, 23(2),113-129. [3] Gutman I,1999. Graph Theory Notes, Newyork, 27,9-15. [4] Mojtaba Shamsipur, Bahram Hemmateenejad, Morteza Akhond, 2004. Highly Correlating Distance/Connectivity-Based Topological Indices, Bull. Korean Chem. Soc. Vol.25, No.2, 253-259. [5] Prabhakara Rao N, Lavanya Lakshmi K, 2008. On the computation of PI,Sz and S indices of tetragonal systems, Acta Ciencia Indica, Vol.XXXIV M, No.4,1753-1766. [6] Prabhakara Rao N, Lavanya Lakshmi K, Laxmi Prasanna A, 2007. On the Wiener index of product Of Path graphs, proceedings of national conference on discrete mathematics and its applications,  NCDMA, Thiagarajar college of engineering, Madurai, p(30)-p(37). [7] Prabhakara Rao N, Lavanya Lakshmi K, Laxmi Prasanna A, 2007.On the Wiener index of Penta chains, proceedings of national conference on discrete mathematics and its applications,  NCDMA, Thiagarajar college of engineering, Madurai. [8] Prabhakara Rao N, Lavanya Lakshmi K, 2008. On PI index of multiple phenylenes, International journal of  combinatorial Graph theory and applications, Vol.1, No.2, pp.117-124. [9] Prabhakara Rao N, Lavanya Lakshmi K, 2009. On Sz index of multiple phenylenes, International Journal of Computational Cognition, Vol.7, No.3, pp.44-49. [10] Prabhakara Rao N, Lavanya Lakshmi K, 2010. On a svd based topological index for polyacenes,International journal of computational cognition, Vol.8, No.4, pp.60-63. [11] Lavanya Lakshmi K, On Wiener index of multiple phenylenes, International journal of combinatorial Graph theory and applications, Vol.2, No.1, pp.57-62. [12] Lavanya Lakshmi K, 1947. On Fr indices of phenylenes, International journal of computational cognition, Vol.8, No.3, pp.44-46, september-2010.Wiener H, Structural Determination of Paraffin boiling points, Journal of  American Society,  69,17-20.     

A highly correlated topological index for polyacenes

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A highly correlated topological index for polyacenes

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