Eccentric Connectivity Index, Hyper and Reverse-Wiener Indices of the Subdivision Graph

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1 Gen. Math. Notes, Vol., No., February 011, pp.4-46 ISSN ; Copyright c ICSRS Publication, Available free online at Eccentric Connectivity Index, Hyper and Reverse-Wiener Indices of the Subdivision Graph Ranjini P.S. 1 and V. Lokesha 1 Department of Mathematics, Don Bosco Institute Of Technology, Kumbalagoud, Bangalore-74, India ranjini p s@yahoo.com Department of Mathematics, Acharya Institute of Technology, Soldevanahalli, Hesaragatta Road, Bangalore-90, India lokeshav@acharya.ac.in Received: /Accepted: Abstract If G is a connected graph with vertex set V, then the eccentric connectivity index of G, ξ (c) (G) is defined as deg(v).ec(v) where deg(v) is the degree of a vertex v and ec(v) is its eccentricity. The Wiener index W (G) = 1[ d(u, v)], the hyper-wiener index W W (G) = 1[ d(u, v) + d (u, v)] and the reverse- Wiener index (G) = n(n 1)D W (G), where d(u, v) is the distance of two vertices u, v in G, d (u, v) = d(u, v), n = V (G) and D is the diameter of G. In this paper, we determine the eccentric connectivity index of the subdivision graph of the complete graphs, tadpole graphs and the wheel graphs. Also, derive an expressions for the hyper and reverse-wiener indices of the same class of graphs. Keywords: Eccentric connectivity index, Wiener index, Hyper-Wiener index, Reverse-Wiener index, Subdivision Graph. 1 Introduction Critical step in pharmaceutical drug design continues to be the identification and optimization of compounds in a rapid and cost effective way. An important tool in this work is the prediction of physico-chemical, pharmacological and

2 Eccentric Connectivity Index, Hyper... 5 toxicological properties of a compound directly from its molecular structure. This analysis is known as the study of the Quantitative Structure Activity relationship (QSAR). In chemistry, a molecular graph represents the topology of a molecule, by considering how the atoms are connected. This can be modelled by a graph, where the points represent the atoms, and the edges symbolize the covalent bonds. Relevant properties of these graph models are studied, giving rise to numerical graph invariants. The parameters derived from this graphtheoretic model of a chemical structure are being used not only in QSAR studies pertaining to molecular design and pharmaceutical drug design, but also in the environmental hazard assessment of chemicals. Many such graph invariant topological indices have been studied. The first, and most well-known parameter, the Wiener index, was introduced in the late 1940 s in an attempt to analyze the chemical properties of paraffins (alkanes) [0]. This is a distance-based index, whose mathematical properties and chemical applications have been widely researched. Numerous other indices have been defined, and more recently, indices such as the eccentric distance sum, and the adjacency- cum-distance-based eccentric connectivity index have been considered [6, 7, 8, 10, 1]. These topological models have been shown to give a high degree of predictability of pharmaceutical properties, and may provide leads for the development of safe and potent anti-hiv compounds. The hyper-wiener index of acyclic graphs was introduced by Randic in 199. Then Klein et al. generalized Randic s definition for all connected graphs, as a generalization of the Wiener index [1]. For the mathematical properties of hyper- Wiener index[9] and its applications in chemistry we refer to [, 4, 9]. The reverse-wiener index[] was proposed by Balaban et al. in 000 [1]; it is important for a reverse problem and applications in modelling of structure property relations. Some mathematical properties of the reverse-wiener index may be found in []. Consider a simple connected graph G, and let V (G) and E(G) denote its vertex and edge sets, respectively. The distance between u and v in V (G), d(u, v) is the length of the shortest u v path in G. The eccentricity, ec(u) of a vertex u ɛ V (G) is the maximum distance between u and any other vertex in G. The diameter D of G, is defined as the maximum value of the eccentricities of the vertices of G. Finally, the degree of a vertex vɛv (G), deg(v) is the number of edges incident to v. We recall definitions which are essential to our work. The eccentric connectivity index ξ c (G) = deg(v).ec(v). The Wiener index W (G) = 1 [ d(u, v)]. The hyper Wiener index W W (G) = 1 [ d(u, v) + d (u, v)]. The reverse Wiener index (G) = n(n 1)D W (G).

3 6 Ranjini P.S. et al. The T n,k Tadpole graph [5, 18] is the graph obtained by joining a cycle graph C n to a path of length k. The wheel graph W n+1 [14] is defined as the graph K 1 + C n, where K 1 is the singleton graph and C n is the cycle graph. The subdivision graph [15, 19] S(G) is the graph obtained from G by replacing each of its edge by a path of length, or equivalently, by inserting an additional vertex into each edge of G [16, 17]. For all terminologies and notations which are not defined in this paper, we refer to F. H. Harary [11]. Eccentric Connectivity Index, Hyper and Reverse-Wiener Indices of the Subdivision Graph In this section, we derived an expression for the eccentric connectivity index, Wiener index, hyper and reverse-wiener indices of the subdivision graphs of the complete graphs, tadpole graphs and the wheel graphs. Theorem.1. The eccentric connectivity index of the subdivision graph of the complete graph { S n is 7n(n 1), for n 4; ξ (c) (S(S n )) = 18, for n=. Proof. The cardinality of the vertex set of the S(S n ) = n(n+1), among which the n vertices are of valency n 1 and the remaining n(n 1) vertices are the subdivision vertices. For the case n = S(S n ) is the cycle C 6, in which each vertex of same eccentricity. Hence ξ (c) (S(S n )) = 18, for the case n =. For n 4, the vertex with degree n 1 is of eccentricity and the subdivision vertices are of eccentricity 4. Hence ξ (c) (S(S n )) = 7n(n 1). Theorem.. The eccentric connectivity index of the subdivision graph of the wheel graph W n+1 is 8n, when n 6. ξ (c) 180, when n=5; (S(W n+1 )) = 14, when n=4; 84, when n=. Proof. The S(W n+1 ) contains the subgraph S(C n ) and the hub of the wheel is of degree n and the n vertices are of degree, and the remaining n vertices are the subdivision vertices. For all values of n, n, the hub of the wheel is of eccentricity and the subdivision vertices on the spokes are of eccentricity 4. The vertex of degree is of eccentricity for the case n = and is of eccentricity 4 for the case n = 4 and is of eccentricity 5 for the case n 5. The subdivision vertices of C n in S(W n+1 ) are of eccentricity 4 for the

4 Eccentric Connectivity Index, Hyper... 7 case n = and n = 4 and are of eccentricity 5 for the case n = 5 and are of eccentricity 6 for the case n 6. Hence, ξ (c) (S(W n+1 )) =.n +.4.n +..n +.4.n = 8n = 84, for the case n =. ξ (c) (S(W n+1 )) =.n +.4.n +.4.n +.4.n = 1n = 14, for the case n = 4. ξ (c) (S(W n+1 )) =.n +.4.n +.5.n +.5.n = 6n = 180, for the case n = 5. ξ (c) (S(W n+1 )) =.n +.4.n +.5.n +.6.n = 8n, for the case n 6. Theorem.. The eccentric connectivity index of the subdivision graph of the tadpole graph T n,k is 1k(k + n), for the case n=k; ξ (c) 1nk + 1k (S(T n,k )) = + 4n + n(n 4k ), for the case n > k; 5 n + 10nk + 6k, when n < k and n even; 5 n + 10nk + 6k + 1, when n < k and n odd. Proof. The S(T n,k ) contains the the subgraph S(C n ) and the path S(P k ). We calculate d(u)e(u) along S(C n ) and along S(P k ). Let v 1 be the vertex in S(C n ) with maximum eccentricity n + k. Then the two neighbors of v 1 say v and v are of eccentricity n + k 1, and the neighbors of v and v are of eccentricity n+k, and so on. Let v l be the vertex of degree in S(T n,k ). For the case n > k, the 4k + 1 vertices in S(C n ) are of eccentricity n + k, n + k 1, n + k 1, n + k, n + k,..., n, n. Then from the remaining n 4k 1 vertices in S(C n ), the n 4k vertices of degree and v l of degree are of eccentricity n. Hence d(u)e(u) in S(C n ) for the case n > k is 8nk + 8k + 5n + n(n 4k ) (1) For the case n = k, the vertex v l of degree is of eccentricity n. d(u)e(u) in S(Cn ) for the case n = k is Hence 8nk + 8k + n () For the case n < k, for any value of n, the vertices in S(C n ) are of eccentricity n+k, n+k 1, n+k 1,..., k +1, k +1 and the vertex v l is of eccentricity k. Hence d(u)e(u) in S(C n ) for the case n < k is n + 8nk + k () Now, we calculate d(u)e(u) along the vertices in the path. Let v l1 e 1 v l e...v lk 1 v l be the path S(P k ) in S(T n,k ). The pendent vertex v l1 is of maximum eccentricity n + k. For the case n k, for all values of n, the vertices of S(P k ) starting from v l1

5 8 Ranjini P.S. et al. is of eccentricity n + k, n + k 1,..., n + 1. Hence d(u)e(u) along the path for the case n k is 4nk + 4k n (4) For the case n < k, e(v) in S(P k ) when n odd, starting from v l1, the first n+k vertices are of eccentricity n+k, n+k 1, n+k,..., n+k and the remaining k n+k vertices are of eccentricity n+k, n+k + 1,..., k 1. Hence d(u)e(u) along the path for the case n < k and n odd is 1 n + nk + 6k k + 1 For the case n < k, e(v) in S(P k ) when n even, starting from v l1, the vertices in S(P k ) are of eccentricity n+k, n+k 1, n+k,..., n+k, n+k +1,..., k 1. Hence d(u)e(u) along the path S(P k ) for the case n < k and n even is (5) 1 n + nk + 6k k (6) Adding equations 1 and 4, the eccentric connectivity index of S(T n,k ) for the case n > k and n even or odd is ξ (c) (S(T n,k )) = 1nk + 1k + 4n + n(n 4k ). Adding equations and 4, the eccentric connectivity index of S(T n,k ) for the case n = k and n even or odd is ξ (c) (S(T n,k )) = 1k(k + n). Adding equations and 5, the eccentric connectivity index of S(T n,k ) for the case n < k and n odd is ξ (c) (S(T n,k )) = 5 n + 10nk + 6k + 1. Adding equations and 6, eccentric connectivity index of S(T n,k ) for the case n < k and n even is ξ (c) (S(T n,k )) = 5 n + 10nk + 6k. Theorem.4. The Wiener index, the hyper Wiener index and the reverse Wiener index of the subdivision graph of the complete graph S n is W (S(S n )) = n (n 1), W W (S(S n )) = n(n 1)(5n 7n+4) and (S(S n )) = n(n+)(n 1). Proof. The cardinality of the vertex set of S(S n ) is n(n+1) among which the n(n 1) vertices are the subdivision vertices and the n vertices are of degree n 1. All the vertices in S n are at a distance in S(S n ). The d(u, v) and d(u, v) among the vertices of S n in S(S n ) is n(n 1) (7) n(n 1) (8)

6 Eccentric Connectivity Index, Hyper... 9 Each vertex of degree n 1 is at a distance 1 with its adjacent n 1 subdivision vertices and at a distance with the remaining (n 1)(n ) subdivision vertices. Hence d(u, v) and d(u, v) among the vertices of degree n 1 and the subdivision vertices is n(n 1)(n 4) (9) n(n 1)(9n 16) (10) Each subdivision vertex is at a distance with the n 4 subdivision vertices and at a distance 4 with the remaining (n )(n ) subdivision vertices. Hence, the d(u, v) and d(u, v) among the subdivision vertices of S n is n(n 1) (n ) n(n 1)(n ) Adding equations 7,a1.9 and 11, The Wiener index of S(S n ) is W (S(S n )) = n (n 1) (11) (1) (1) Adding equations 8,10 and 1, d(u, v) among all the vertices of S(S n ) is n(n 1)(4n 7n + 4) (14) From equation 1 and 14 the hyper wiener index of S(S n ) is W W (S(S n )) = n(n 1)(5n 7n+4). The diameter of S(S n ) is 4 with which the reverse wiener index is (S(S n )) = n(n+)(n 1). Theorem.5. The Wiener index, the hyper Wiener index and the reverse Wiener index of the subdivision graph of the wheel graph W n+1 is n(9n 1), for the case n 5; 188, for the case n=4; W (S(W n+1 )) = 96, for the case n=. W W (S(W n+1 )) = 4n(4n 9), for the case n 5; 716, for the case n=4; 6, for the case n=. and

7 40 Ranjini P.S. et al. (S(W n+1 )) = n(9n + 5), for the case n 6; 80, for the case n=5; 14, for the case n=4; 84, for the case n=. Proof. In S(W n+1 ), a vertex of degree is at a distance 1,, with the neighboring subdivision vertices on a spoke, to the hub and to the remaining n 1 subdivision vertices on the remaining spokes. Hence the d(u, v) and d(u, v) from the vertices of degree to the subdivision vertices on the spokes and to the hub is respectively as n (15) n(9n 4) (16) Also the vertex of degree is at a distance with the two neighboring vertices of degree and at a distance 4 with the remaining n vertices of degree. So d(u, v) and d(u, v) among the vertices of degree is n(n ) (17) 4n(n 5) (18) Also d(u, v) and d(u, v) from the vertex of degree to the subdivision vertices of C n for the case n = is 5n (19) 11n (0) For the case n 4, the vertex of degree is at a distance 1 with the two neighboring subdivision vertices vertices of C n and at a distance with two more neighboring subdivision vertices of C n and at a distance 5 with the remaining n 4 subdivision vertices of C n. So d(u, v) and d(u, v) from the vertex of degree to the subdivision vertices of C n for the case n 4 is n(5n 1) (1) n(5n 16) () The subdivision vertices of C n are at a distance with the neighboring subdivision vertices on the spoke and at a distance with the hub and a distance 4 with the remaining n subdivision vertices on the spoke. Hence d(u, v) and d(u, v) from the subdivision vertices of C n to the vertices on the spoke and to the hub for all values of n is n(4n 1) ()

8 Eccentric Connectivity Index, Hyper n(16n 15) (4) For the case n =, the subdivision vertices of C n are at a distance with each other. Hence d(u, v) and d(u, v) among the subdivision vertices of C n for the case n = is n (5) 4n (6) For the case n = 4, the the subdivision vertices of C n are at a distance,, 4 with each other. Hence the d(u, v) and d(u, v) among the subdivision vertices of C n for the case n = 4 is 4n (7) 1n (8) For the case n 5, a subdivision vertex of C n are at a distance,, 4, 4 and 6 respectively with the four adjacent subdivision vertices and remaining n 5 subdivision vertices of C n. Hence d(u, v) and d(u, v) among the subdivision vertices of C n for the case n 5 is n(n ) (9) n(9n 5) (0) Now to calculate the d(u, v) and d(u, v) among the subdivision vertices on the spoke and to the hub, the distance between a pair of subdivision vertices on the spoke is and with the hub is 1. Hence the d(u, v) and d(u, v) among the subdivision vertices on the spoke and to the hub is n (1) n(n 1) () Hence adding equations 16, 18, 0, 4, 6 and, the Wiener index of the subdivision graph of the wheel graph is W (S(W n+1 )) = n(5n + 1) = 96, for the case n = () Adding the equations 16, 18,, 4, 8 and, the Wiener index of the subdivision graph of the wheel graph is W (S(W n+1 )) = n(15n 1) = 188, for the case n = 4 (4) Adding the equations 16, 18,, 4, 0 and, the Wiener index of the subdivision graph of the wheel graph is W (S(W n+1 )) = n(9n 1), for the case n 5 (5)

9 4 Ranjini P.S. et al. Adding the equations 17, 19, 1, 5, 7 and, d(u, v) when n = is 5n(7n 5), for the case n = (6) Adding the equations 17, 19,, ( 5, 9 and, d(u, v) when n = 4 is 1n(5n 9), for the case n =4 (7) Adding equations 17, 19,, ( 5, 1 and, d(u, v) when n = 4 is n(9n 95) for the case n 5 (8) From equations 4 and 8, the hyper wiener index of S(W n+1 ) for the case n = is W W (S(W n+1 )) = n(45n ) = 6. From equations 5 and 8, the hyper wiener index of S(W n+1 ) for the case n = 4 is W W (S(W n+1 )) = n(75n 11) = 716. From equations 6 and 8, the hyper wiener index of S(W n+1 )) for the case n 5 is W W (S(W n+1 )) = 4n(4n 9). When n =, 4, 5 the diameter of S(W n+1 ) is 4, 4 and 5 respectively and for all values of n 6 the diameter is 6. The cardinality of the vertex set of S(W n+1 ) is n + 1 for all n. Hence from equation (.), (S(W n+1 )) = 4n(n + 1) = 84, for the case n =. From equation 5, (S(W n+1 )) = n(n + 19) = 14, for the case n = 4. From equation 6, (S(W n+1 )) = n(9n+67) = 80, for the case n = 5. From equation 6, with the additional concept the diameter of S(W n+1 ) is 6 for n 6 (S(W n+1 )) = n(9n + 5), for the case n 6. Theorem.6. The Wiener index, the hyper Wiener index and the reverse Wiener index of the subdivision graph of the tadpole graph T n,k is W (S(T n,k )) = (4k 1)k + kn + n(n+1) + kn(k + n) + n, W W (S(T n,k ) = (4k k+10nk+n+4n +4k 4 k +16k n+4kn +n 4 ) + (n + 8k n + 4n k + 4n k ) and (S(T n,k )) = n (k 1 n) + k (n 1 + 4k) + ( 4nk + k n). Proof. The S(T n,k ) contains a path S(P k ) of length k and a cycle S(C n ). Let v l1 be the unique vertex of degree in S(T n,k ) and let v l1 e l1 v l e l...e lk v lk+1 be the path S(P k ) of length k. Starting from v l1, the vertices in S(P k ) are at a distance 1,,,..., k. From v l, the remaining vertices in S(P k ) are at a

10 Eccentric Connectivity Index, Hyper... 4 distance 1,,,..., k 1. Proceeding like this the vertex v lk is at a distance 1 with v lk+1. Hence d(u, v) and d(u, v) along the path S(P k ) is k(k + 1)(k + 1) (9) k(k + 1)(k + 1) (40) Let P 1 = v 1 e 1 v e...v n 1 e n v l1 e l1 v l e l...e lk v lk+1 be the largest path in S(T n,k ) of length n + k. Then P = v e...v n 1 e n v l1 e l1 v l e l...e lk v lk+1 and P = v e...v n 1 e n v l1 e l1 v l e l...e lk v lk+1 be the two paths in S(T n,k ) of length n + k 1. P = v e...v n 1 e n v l1 e l1 v l e l...e lk v lk+1 and P = v e...v n 1 e n v l1 e l1 v l e l...e lk v lk+1 be the paths in S(T n,k ) of length n + k. Continuing like this P n = v n 1 e n v l1 e l1 v l e l...e lk v lk+1 and P n = v n 1 e n v l1 e l1 v l e l...e lk v lk+1 be the paths in S(T n,k ) of length k + 1. Hence d(u, v) and d(u, v) along P 1 is (k + n)(k + n + 1) (41) (k + n + 1) d(u, v) and d(u, v) along P and P (k + n + 1) + k + n (4) (k + n)(k + n 1) (k + n) (k + n) + k + n Proceeding like this d(u, v) and d(u, v) along P n and P n (k + ) (k + 1)(k + ) (k + ) + k + 1 Adding equations 41, a1.4,..., a1.45, d(u, v) along P 1, P, P...P n, P n k + kn k + n + n 6 + 4k n + kn + n (4) (44) (45) (46) is (47)

11 44 Ranjini P.S. et al. Adding equations 4,,..., 46, d(u, v) along P 1, P, P...P n, P n is (4nk k + n + n )(8k + 6k nk + n + n ) 6 (48) For an even cycle with cardinality n of the vertex set V the W iener index W (C n ) = n and d(u, v) = n (n +). Hence for S(C 8 4 n ) in S(T n,k ) W iener index and d(u, v) is n (49) n (4n + ) 6 d(u, v) and d(u, v) from v to v l1, v to v l1,...,v n 1 to v l1 is (50) n(n 1) (51) n 4 6 n (5) 6 Adding equations 9,a1.47, a1.49 and subtracting equation 51, the Wiener index W (S(T n,k )) = (4k 1)k + kn + n(n + 1) + kn(k + n) + n (5) Adding equations 40, 48, 50 and subtracting equation 5, d(u, v) in S(T n,k ) d(u, v) = 4k4 k n +16k +4k n+ 4kn +4n k +kn + 4kn +n 6 +5n 6 +n +n4 (54) Adding equations 5 and??, the hyper wiener index W W (S(T n,k ) = (4k k+10nk+n+4n +4k 4 k +16k n+4kn +n 4 ) + (n + 8k n + 4n k + 4n k ). The cardinality of the vertex set of S(T n,k ) is (n + k) and the diameter is n + k. Hence from equation 5, the reverse wiener index of S(T n,k ) is (S(T n,k )) = n (k 1 n ) + k (n 1 + 4k ) + ( 4nk + k n ). References [1] A.T. Balaban, D. Mills, O. Ivanciuc and S.C. Basak, Reverse Wiener indices, Croat. Chem. Acta., 7 (000), [] X. Cai and B. Zhou, Reverse Wiener indices of connected graphs, MATCH Commun. Math. Comput. Chem. 60 (008),

12 Eccentric Connectivity Index, Hyper [] G.G. Cash, Polynomial expressions for the hyper-wiener index of extended hydrocarbon networks, Comput. Chem., 5 (001), [4] G.G. Cash, Relationship between the Hosoya polynomial and the hyper- Wiener index, Appl. Math. Lett., 15 (00), [5] D.M. Cvetkocic, M. Doob and H. Sachs, Spectra of Graphs-Theory and Application Graph classes : A Survey, PA : SIAM, Philadelphia, 19 (1987). [6] H. Dureja and A.K. Madan, Topochemical models for prediction of cyclindependent kinase inhibitory activity of indole--ones, Journal of Molecular Modeling,11 (005), [7] H. Dureja and A.K. Madan, Topochemical models for the prediction of permeability through blood-brain barrier, International Journal of Pharmaceutics, (006), 7-. [8] H. Dureja and A.K. Madan, Predicting anti-hiv activity of dimethylaminopyridin--ones: Computational approach using topochemical descriptors, Chemical Biology and Drug Design, 7 (009), [9] I. Gutman, Relation between hyper-wiener and Wiener index, Chem. Phys. Lett., 64 (00), [10] S. Gupta, M. Singh and A.K. Madan, Application of graph theory: Relationship of eccentric connectivity index and Wiener s index with antiinflammatory activity, Journal of Mathematical Analysis and Applications, 66 (00), [11] F. Harary, Graph Theory, Reading, MA: Addision-Wesley, 46(1994). [1] D.J. Klein, I. Lukovits and I. Gutman, On the definition of the hyper- Wiener index for cycle-containing structures, J. Chem. Inf. Comput. Sci., 5 (1995), [1] M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi and S.G. Wagner, Some new results on distance-based graph invariants, European Journal of Combinactorics, 0 (009), [14] S. Pemmaraju and S. Skiena, Cycles, Stars and Wheels 6..4 in Computational Discrete Mathematics, Graph Theory in Mathematica, Cambridge University Press, Cambridge, England. [15] P.S. Ranjini, V. Lokesha and M.A. Rajan, On Zagreb indices of the subdivision graphs, Int. J. Math. Sc. Eng. Appl., 4(IV) (010), 1-8.

13 46 Ranjini P.S. et al. [16] P.S. Ranjini and V. Lokesha, The Smarandache-Zagreb indices on the three graph operators, Int. J. Math. Combin., China, (010), [17] P.S. Ranjini and V. Lokesha, On the Szeged index and the PI index of the certain class of Subdivision graphs, Bulletin of Pure and Applied Mathematics, 6(1) (01). [18] E.W. Weisstein, Tadpole Graph, Math world-a Wolfram Web Resource. [19] W. Yan, Bo-Yin Yang and Yeong-Nan Yeh, Wiener Indices and Polynomials of Five Graph Operators, Academia Sinica Taipei, Taiwan. [0] H. Wiener, Structural determination of the paraffin boiling points, Journal of the American Chemical Society, 69 (1947), [1] B.Y. Yang and Y.N. Yeh, Y. Fong et al eds, About Wiener numbers and polynomials, Proc. Of Second International Taiwan-Moscow Algebra workshop (Tainan, 1997), Springer(Hong Kong), (000), 0-6.