Superaugmented Pendentic Indices : Novel Topological Descriptors for QSAR / QSPR

Four pendenticity based topological descriptors termed as superaugmented pendentic indices have been conceptualized in the present study. An in-house computer program was utilized to compute index values of all the possible structures (with at least one pendent vertex) containing four, five and six vertices. The sensitivity towards branching, discriminating power, degeneracy and mathematical properties of the proposed superaugmented pendentic indices were investigated. All the four proposed indices exhibited exceptionally high sensitivity towards branching, high discriminating power and extremely low degeneracy. Superaugmented pendentic index-4 (∫-4) exhibited exceptionally high discriminating power of 114 in structures containing only six vertices. Statistical significance of the proposed indices was investigated using intercorrelation analysis with Wiener’s index, Balaban’s mean square distance index, molecular connectivity index, Zagreb indices (M1 and M2), superpendentic index and eccentric connectivity index. The exceptionally high sensitivity towards branching, high discriminating power amalgamated with extremely low degeneracy offer proposed indices a vast potential for isomer discrimination, similarity/dissimilarity, drug design, quantitative structureactivity/structure-property relationships, lead optimization and combinatorial library design.


Introduction
In the recent years, "Graph Theory" has been applied in an automated computer treatment of chemical structures and QSAR [1].In chemical graph theory, molecular structures are represented by hydrogen suppressed graphs (commonly known as molecular graphs) in which vertices represent the atoms and bonds are represented by edges.The connection between atoms can be described by various types of topological matrices, which can be mathematically manipulated so as to derive a single number, usually known as graph invariant, graph-theoretical index or topological index (TI) [2].Topological indices (TIs) are widely used as structural descriptors in QSPR/QSAR models [3].Moreover, novel strategies are being adopted for continuous search and development of TIs [4].During recent years, significant progress has been reported in the development of various topological, geometric, electrostatic, and quantum chemical indices as molecular descriptors.Because of the simplicity of topological structural representation, these TIs are sometime preferred over more complex geometric, electrostatic, and quantum chemical descriptors, especially in those cases where their use significantly reduces the computation time [5].The pharmaceutical industry contributed towards increased interest in molecular descriptors because of the necessity to reduce the expenditure involved in synthesis, in vitro, in vivo and clinical testing of new medicinal compounds [6].In recent years, a large number of topological indices have been reported and utilized for chemical documentation, isomer discrimination, study of molecular complexity, chirality, similarity/dissimilarity, QSAR/QSPR, drug design and database selection, lead optimization, rational combinatorial library design and for deriving multilinear regression models [2,7,8].Estrada defined the paradigm of the use of TIs and molecular descriptors in general in QSAR studies as "it is desired to have as many molecular descriptor as possible at our disposition, but it is preferred to include as few of them as possible in the QSPR and QSAR models to be developed" [3].
In the present investigation, four pendenticity based topological descriptors termed as superaugmented pendentic indices, denoted by SA ∫ P have been conceptualized and their mathematical properties studied.The sensitivity towards branching, discriminating power, degeneracy and intercorrelation of the proposed indices with regard to all the possible structures containing four, five and six vertices (with at least one pendent vertex) have been investigated.

Results and Discussion
Topological descriptors have gained considerable popularity as these can be derived from molecular structures using low computational resources [9].The use of TIs in the design and selection of novel active compounds is probably one of the most active areas of research in the application of such descriptors to biological problems [8].In recent years a large number of topological indices have been reported and utilized for chirality, similarity/dissimilarity, QSAR/QSPR, drug design and database selection, lead optimization and for rational combinatorial library design.Though a large number of molecular descriptors of diverse nature have been reported in literature but only a small proportion of these descriptors have been successfully utilized in QSAR.As a consequence there is strong need to develop non-correlating topological indices with sensitivity towards branching, high discriminating power and extremely low degeneracy.

Tab. 1.
Index values of superaugmented pendentic indices for all possible structures of four, five and six vertices containing at least one pendent vertex.

SA
∫ P- In the present study, four superaugmented pendentic indices, denoted by SA ∫ P , were conceptualized.These indices can be easily calculated from pendent matrix (Dp) and additive adjacency matrix (A α ).In the proposed TIs, simultaneous use of pendent-distance, degree of the vertices and eccentricity results in significant changes in the index value with a minor change in the branching of molecules.
As observed from Figure 1, the value of superaugmented pendentic index-1 ( SA ∫ P-1 ) changes by more than 4 times (from 16.0 to 65.0), the value of superaugmented pendentic index-2 ( SA ∫ P-2 ) changes by more than 5 times (from 6.333 to 33.0) and the value of superaugmented pendentic index-3 ( SA ∫ P-3 ) changes by more than 6 times (from 2.694 to 17.0) and the value of superaugmented pendentic index-4 ( SA ∫ P-4 ) changes by more than 7 times (1.211 to 9.0) following branching of five membered linear carbon structure.The superaugmented pendentic index-4 was found to be about 2 times more sensitive to change in molecular structure when compared with superaugmented pendentic index-1 for identical changes.These superaugmented pendentic indices were found to be far more sensitive towards branching using three isomers of pentane.
Researchers are striving hard to develop TIs with not only high discriminating power but also devoid of both degeneracy and correlation with existing TIs.The values of superaugmented pendentic indices were computed for all the possible structure of four, five and six vertices containing at least one pendent vertex using an in-house computer program.Various structures containing four, five and six vertices containing at least one pendent vertex and their corresponding index values have been presented in Table 1 whereas their comparison has been depicted in Table 2. Superaugmented pendentic indices have revealed high discriminating power.The discriminating power may be defined as the ratio of highest to lowest value for all possible structures of same number of vertices.The ratio of highest to lowest value for all possible structure containing six vertices with at least one pendent vertex was found to be 31 in case of superaugmented pendentic index-1, 46 in case of superaugmented pendentic index-2, 73 for superaugmented pendentic index-3 and 114 for superaugmented pendentic index-4 in comparison to 2.83 for ∫ P .High discriminating power of the proposed indices renders them more sensitive to any change(s) in the molecular structure.Extreme sensitivity towards branching as well as exceptionally high discriminating power of all the four proposed indices is clearly evident from the respective index values (Table 1) of all the possible structures with four, five and six vertices containing at least one pendent vertex.
Degeneracy is the measure of ability of an index to differentiate between the relative positions of atom in a molecule.The superaugmented pendentic index-2, superaugmented pendentic index-3 and superaugmented pendentic index-4 did not exhibit any degeneracy for all possible structures with six vertices containing at least one pendent vertex, whereas the superaugmented pendentic index-1 has very low degeneracy of five in case of all possible structures with six vertices containing at least one pendent vertex whereas The ∫ P had 22 same values out of 39 structures with all possible structures with six vertices containing at least one pendent vertex (Table 2).Extremely low degeneracy indicates the enhanced capability of these indices to differentiate and demonstrate slight variations in the molecular structure.This means that the likeliness of different structures to have same value is remote.Sci Pharm.2009; 77; 521-537.
Intercorrelation analysis of the proposed four indices with other well-known indices revealed that superaugmented pendentic indices are not correlated with Wiener's index [10], Balaban's mean square distance index [11], molecular connectivity index [12] and eccentric connectivity index [13] and superpendentic index [14] as well.Moreover, superaugmented pendentic indices are weakly correlated with Zagreb indices (M 1 and M 2 ) [15,16].These superaugmented pendentic indices describe the structural parameters in a different manner in comparison to other indices.
Throughout, let G = (V, E) be a simple connected graph with vertex set V = {v 1 ,v 2 ,……,v n } and edge set E. Let d i be the degree of vertex v i for i = 1, 2, …..,n and Δ, the highest degree of a graph G. Denote by i ~ j, vertices v i and v j are adjacent.For two vertices v i and v j (i ≠ j), d (i, j), the topological distance between v i and v j is the number of edges in a shortest path joining v i and v j .The diameter of a graph is the maximum distance between any two vertices of G.The graphs having at least one pendent vertex were denoted by G k,n , where k (k ≥ 1) is the number of pendent vertices and n (> 4) is the order of the graph.

Superpendentic index (∫ p ):
The superpendentic index, denoted by ∫ P , is defined as square root of the sum of products of non-zero row elements in the pendent matrix in the hydrogen suppressed molecular graph [14].It is expressed as, ( ) ( ) where m and n are maximum possible number of i and j respectively.For a molecular graph (G) if v 1 , v 2 ,….., v n are its vertices.Then, the topological distance d(v i ,v j ⏐G) between the vertices v i and v j of G is length of the shortest path connecting v i with v j .
A pendent vertex is defined as a vertex of degree one or an endpoint.The eccentricity e i of a vertex v i in G is the length of shortest path from v i to the vertex v j that is farthest from v i (e i = max d (v i , v j ; j / G) [16].

Superaugmented pendentic index-1:
The superaugmented pendentic index-1, denoted by SA ∫ P-1 , is defined as the summation of quotients, of the product of non-zero row elements in the pendent matrix and product of adjacent vertex degrees; and eccentricity of the concerned vertex, for all vertices in the hydrogen suppressed molecular graph.It is expressed as, , ) (G where p i = ∏ j: dj =1; p ij≠ 0 p ij , m i = ∏ j:j~i d j , and p ij is the length of the path that contains the least number of edges between vertex v i and vertex v j in graph G k,n , m i is the product of degrees of all the vertices (v j ), adjacent to vertex i and can be easily obtained by multiplying all the non-zero row elements in additive adjacency matrix, d i is the degree of the vertex v i , e i is the eccentricity of vertex v i .

Superaugmented pendentic index-2:
The superaugmented pendentic index-2, denoted by SA ∫ P-2 , is defined as the summation of quotients, of the product of non-zero row elements in the pendent matrix and product of adjacent vertex degrees; and squared eccentricity of the concerned vertex, for all vertices in the hydrogen suppressed molecular graph.It is expressed as where p i = ∏ j: dj =1; p ij≠ 0 p ij , m i = ∏ j:j~i d j , and p ij is the length of the path that contains the least number of edges between vertex v i and vertex v j in graph G k,n , m i is the product of degrees of all the vertices (v j ), adjacent to vertex i and can be easily obtained by Sci Pharm.2009; 77; 521-537.multiplying all the non-zero row elements in additive adjacency matrix, d i is the degree of the vertex v i , e i is the eccentricity of vertex v i .

Superaugmented pendentic index-3:
The superaugmented pendentic index-3, denoted by SA ∫ P-3 , is defined as the summation of quotients, of the product of non-zero row elements in the pendent matrix and product of adjacent vertex degrees; and cubic eccentricity of the concerned vertex, for all vertices in the hydrogen suppressed molecular graph.It is expressed as, , ) (G where p i = ∏ j:dj =1; p ij≠ 0 p ij , m i = ∏ j:j~i d j , and p ij is the length of the path that contains the least number of edges between vertex v i and vertex v j in graph G k,n , m i is the product of degrees of all the vertices (v j ), adjacent to vertex i and can be easily obtained by multiplying all the non-zero row elements in additive adjacency matrix, d i is the degree of the vertex v i , e i is the eccentricity of vertex v i .

Superaugmented pendentic index-4:
The superaugmented pendentic index-4, denoted by SA ∫ P-4 , is defined as the summation of quotients, of the product of non-zero row elements in the pendent matrix and product of adjacent vertex degrees; and fourth power of eccentricity of the concerned vertex, for all vertices in the hydrogen suppressed molecular graph.It is expressed as, , ) (G where p i = ∏ j:dj =1; p ij≠ 0 p ij , m i = ∏ j:j~i d j , and p ij is the length of the path that contains the least number of edges between vertex v i and vertex v j in graph G k,n , m i is the product of degrees of all the vertices (v j ), adjacent to vertex i and can be easily obtained by multiplying all the non-zero row elements in additive adjacency matrix, d i is the degree of the vertex v i , e i is the eccentricity of vertex v i .
Superaugmented pendentic indices ( SA ∫ P ) can be easily calculated from pendent matrix (Dp) and additive adjacency matrix (A α ) obtained by modifying distance matrix (D) and adjacency matrix (A), respectively.Pendent matrix (Dp) of a graph G is a sub-matrix of distance matrix (D) obtained by retaining the columns corresponding to pendent vertices.The additive adjacency matrix (A α ) is obtained from adjacency matrix by substituting the degree of corresponding vertex (of the vertices adjacent to vertex i) of a molecular graph G.The product of the non-zero row elements in additive adjacency matrix represents the m i .Calculation of superaugmented pendentic index-1 ( SA ∫ P-1 ), superaugmented pendentic index-2 ( SA ∫ P-2 ), superaugmented pendentic index-3 ( SA ∫ P-3 ) and superaugmented pendentic index-4 ( SA ∫ P-4 ) for three isomer of pentane has been exemplified in Fig. 1.
The sensitivity of superaugmented pendentic indices to branching was investigated using three isomers of pentane (Fig. 1).Discriminating power and degeneracy of the superaugmented pendentic indices were investigated using all possible structures with four, five and six vertices containing at least one pendent vertex (Table 1) and compared for discriminating power and degeneracy (Table 2).
The intercorrelation of four proposed superaugmented pendentic indices with Wiener's index (W), Balaban's mean square distance index (D), molecular connectivity index (χ), Zagreb indices (M 1 and M 2 ), superpendentic index (∫ P ), and eccentric connectivity index (ξ c ) was investigated (Table 3).This intercorrelation has been determined with respect to index values of all possible structures containing four, five and six vertices (with at least one pendent vertex).The degree of correlation was appraised by the correlation coefficient r.Pairs of indices with r ≥ 0.97 are considered highly inter-correlated, those with 0.90 ≤ r ≤ 0.97 are appreciably correlated, those with 0.50 ≤ r ≤ 0.89 are weakly correlated and finally the pairs of indices with low r-values (< 0.50) are not inter-correlated [17].

Properties on Superaugmented Pendentic Indices of Graphs
Here we study superaugmented pendentic index-1 of graph G k,n , denoted by SA ∫ P-1 (G k,n ) and defined as , ) where p i = ∏ j:dj=1; p ij≠ 0 p ij , m i = ∏ j:j~i d j , and p ij is the length of the path that contains the least number of edges between vertex v i and vertex v j in graph G k,n , d i is the degree of the vertex v i , e i is the eccentricity of vertex v i .
Denote star K 1, n-1 , and path P n .Now we calculate SA ∫ P-1 for star K 1, n-1 , and path P n (n > 3):   Suppose that vertices v i , i=1, 2, …….,k are pendent vertices and vertex v n is the highest degree vertex corresponding degree Δ in G k.n .Let v1 be only one pendent vertex in G 1,n , and we define p 1 = 1 in G 1, n .Denote by H 1,n , which is constructed by the complete graph K n-1 of order n-1 with one pendent vertex.Now we calculate (H 1,n ) → SA ∫ P-1 (H 1,n ): where μ is the maximum average degree of all non-pendent vertices and D is the diameter of T.Moreover, the equality holds in (6) if and only if T is a star K 1, n-1 .
Proof: First let v i , i=1, 2,……,k be the pendent vertices of tree T. Also let μ i be the average degree of the vertices adjacent to vertex v i , that is, and let μ be the maximum average degree of all non-pendent vertices.For i = k + 1, k + 2,……..,n; , : For any i, we have D ≥ e i ≥ d(i, j) for all v j .

Using this we have
For i = k + 1, k + 2, ……….,n; we have We have 8) and ( 9) First part of the proof is over.Now suppose that equality in (6) holds.Then all inequalities in the above argument must be equalities.From equality in ( 7) and ( 10), we get: for each non-pendent vertex v i , d l = d k for all k, l such that k ~ i, l ~ i and d i = Δ, μ i =μ.
We can see easily that for any non-pendent vertex v i , d(i, j) ≤ D-1 (v j is any vertex) in any tree.From equality in (9), we must have d(i, j) = D -1 for any non-pendent vertex v i and any pendent vertex v j .
Since T is a tree, from above result we conclude that k = n -1 and hence T is star K 1, n-1 .
Conversely, one can see easily that the equality holds in (6) for star K 1, n-1 .
Corollary 1.3.Let T (≠ P n ) be a tree of order n with k pendent vertices.Then where μ is the maximum average degree of all non-pendent vertices and D is the diameter of T.
Proof: Since tree T is not a path P n , we have that at least three pendent vertices in T. Thus we have For i = k+1, k+2,…….,n;we have Sci Pharm.2009; 77; 521-537.
We have , (T) Using ( 12), ( 13) in ( 14) and from Theorem 1.2, we get the required result.Remark 1.4.In Theorem 1.2, we have that star K 1, n-1 is the maximum superaugmented pendentic index-I of trees.Also we believe that path P n is the minimum superaugmented pendentic index-I of trees.Now we will see that H 1,n is the maximum superaugmented pendentic index-1 of G k,n graphs (k ≥1, n > 4).Proof: We consider two cases (i) k = 1, (ii) k ≥ 2.

Theorem 1 . 5 .
Let G k,n (k ≥ 1, n > 4) be a connected simple graph of order n with k pendent vertices.Then ( ) with equality holding if and only if G k, n ≅ H 1, n .