iscrete Applied Mathematics 57 009 68 633 Contents lists available at Scienceirect iscrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam On reciprocal complementary Wiener number Bo Zhou a,, Xiaochun Cai a, Nenad Trinajstić b a epartment of Mathematics, South China Normal University, Guangzhou 5063, China b The Rugjer Bošković Institute, P. O. Box 80, HR-0 00 Zagreb, Croatia a r t i c l e i n f o a b s t r a c t Article history: Received 3 March 008 Received in revised form 30 June 008 Accepted 8 September 008 Available online 9 October 008 We report properties, especially upper and lower bounds and the Nordhaus Gaddum-type result for the reciprocal complementary Wiener number of a connected molecular graph. 008 Elsevier B.V. All rights reserved. Keywords: Connected graphs Reciprocal complementary Wiener number Upper and lower bounds Nordhaus Gaddum-type result. Introduction The Wiener number []a often also called the Wiener index [] and the related molecular descriptors have a long history [8,5,7,3] since 97 when Harry Wiener 9 998 [9] introduced his number as the path number []. The empirical Wiener s definition of his number has been formalized via the distance matrix []a by Hosoya [6]. In the large family of the Wiener-like molecular descriptors [0], the complementary Wiener number and the reciprocal complementary Wiener number are recent additions. They have been introduced by Ivanciuc [7] and discussed by Ivanciuc et al. [8,9]. Related work may be found in, e.g., [0]. We consider simple molecular graphs, i.e., graphs without multiple edges and loops []b, [3]a. Let G be a connected graph with the vertex-set VG = {v, v,..., v n }. The distance matrix of G is an n n matrix d ij such that d ij is just the distance i.e., the number of edges of a shortest path between the vertices v i and v j in G []a, denoted by dv i, v j G. The complementary distance matrix C of G is an n n matrix c ij such that c ij = d ij if i j, and 0 otherwise []b, where is the diameter of the graph G [3]b. The reciprocal complementary distance matrix RC of G is an n n matrix rc ij such that rc ij = if i c ij j, and 0 otherwise []c. The Hosoya definition of the Wiener number of G, denoted by WG, is given by [6] WG = n n d ij = d ij. i<j i= j= The reciprocal complementary Wiener RCW number of the graph G is similarly defined as [7] RCWG = n n rc ij = rc ij. i<j i= j= Corresponding author. E-mail addresses: zhoubo@scnu.edu.cn B. Zhou, trina@irb.hr N. Trinajstić. 066-8X/$ see front matter 008 Elsevier B.V. All rights reserved. doi:0.06/j.dam.008.09.00
B. Zhou et al. / iscrete Applied Mathematics 57 009 68 633 69 The RCW number has been successfully applied in the structure-property modeling of the molar hear capacity, standard Gibbs energy of formation and vaporization enthalpy of 3 alkanes C 6 C 0 [7]. In the present report, we give some properties, especially various upper and lower bounds and the Nordhaus Gaddumtype result [] of the reciprocal complementary Wiener number.. Bounds for the reciprocal complementary Wiener number Let P n and S n be respectively the path and the star with n vertices. Let K n be the complete graph with n vertices. Let dg, k be the number of the unordered pairs of vertices of G that are at distance k, k =,,...,. Then RCWG = k= dg, k k. We first give upper bounds for RCW number using the graph-parameters such as the number of vertices and the number of edges. Proposition. Let G be a connected graph with n vertices. Then nn RCWG with equality if and only if G = K n. Proof. For i, j =,,..., n with i j, it is easily seen that rc ij = d ij with equality if and only if = d ij =. Thus RCWG nn with equality if and only if =, i.e., G = K n. Proposition. Let G be a noncomplete connected graph with n 3 vertices and m edges. Then nn RCWG m with equality if and only G has diameter. Proof. Note that dg, = nn m and k= dg, k = m. Since G K n, we have, and then RCWG = k= m dg, k k nn with equality if and only if =. dg, dg, k k= nn m = m Let G be a connected graph with n vertices, m edges and diameter. Then by the above argument, nn m RCWG with equality if and only if =. Corollary 3. Let G be a noncomplete connected graph with n 3 vertices. Then n RCWG with equality if and only G = S n. Proof. Let m be the number of G. Then m n with equality if and only if G is a tree. Now the result follows from Proposition. By Corollary 3, if G is a tree with n 3 vertices, then RCWG n with equality if and only if G = S n. Similarly to the argument in Corollary 3, if G is a unicyclic graph with n vertices then RCWG nn with equality if and only if G is the quadrangle, or the pentagon, or the graph formed by attaching n 3 pendant vertices to a vertex of a triangle. For v i VG, Γ v i denotes the set of its first neighbors in G and the degree of v i is δ i = Γ v n i. The term i= δ i is known as the first Zagreb index of G, denoted by M G [,,3,6].
630 B. Zhou et al. / iscrete Applied Mathematics 57 009 68 633 Proposition. Let G be a triangle- and quadrangle-free connected graph with n vertices and m edges. If the diameter of G is at least three, then nn RCWG m 6 M G with equality if and only if G has diameter 3. Proof. Note that there are dg, = m. Since G is triangle- and quadrangle-free, we have dg, = M G m see [6], nn and then k=3 dg, k = M G. Then dg, k dg, dg, RCWG dg, k k k= k=3 m 3 [ ] M nn G m M G nn = m 6 M G with equality if and only if = 3. Now we give lower bounds for RCW number. For a graph G, G stands for its complement []c. A graph is said to be diameter-maximal if the diameter of Ge the graph formed from G by adding the edge e is smaller than that of G for every edge e of G. Lemma 5 [5]. A graph G of diameter is diameter-maximal if and only if there is a vertex v, such that the distance layers V i, where V i = {x dv, x G = i} for i = 0,,...,, fulfill the condition that the subgraph induced by V i V i is complete for any i =,,...,, and if then V =. By similar argument as in [6], we have Proposition 6. Let G be a connected graph with n vertices and diameter. Then RCWG f n, with equality if and only if G is a diameter-maximal graph for which all noncentral layers are trivial, where n n n for even, i f n, = n n n for odd. i Proof. Let G be a graph with minimum RCW number in the class of graphs with n vertices and diameter. Evidently, G is a diameter-maximal graph, and thus it has the form given in Lemma 5. Suppose that there is a noncentral layer V k with V k > and x V k. If k <, then we choose the least such k, and for a new diameter-maximal graph G with layers V 0, V,..., V k, V k {x}, V k {x}, V k,..., V, it is easily seen that RCWG RCWG = k i= i i = k k V ik i= k k = k i= k k i= V ik i i i i = k k < 0, i i
B. Zhou et al. / iscrete Applied Mathematics 57 009 68 633 63 from which we have RCWG < RCWG, a contradiction. If k >, then we choose the maximal such k,and for a new diameter-maximal graph G with layers V 0, V,..., V k, V k {x}, V k {x}, V k,..., V, it is easily seen that k RCWG RCWG = i k V k i i i i i= i= = k k V k i i i i= k k i i i= = k = k k k < 0, a contradiction again. Now we have proved that for G, all noncentral layers are trivial. It may be checked that RCWG = f n,. Proposition 7. Let G be a connected graph with n vertices. Then RCWG n with equality if and only if G = P n. Proof. Let be the diameter of G. Suppose that < n. If is even, then f n, f n, = n i n If is odd, then n n i = n i n n n f n, f n, = n n n n n n [ ] n = n n > 0. n = n n i i n n i n n n n n > 0. Thus, f n, is decreasing for n. Now the result follows from Lemma 6. Let G be a tree with n 3 vertices. By Proposition 7, RCWG n with equality if and only if G = P n. A direct reasoning is as follows. Let w 0 w... w be a diametrical path in G. Suppose that < n. Then for some k =,,...,, w k
63 B. Zhou et al. / iscrete Applied Mathematics 57 009 68 633 has a neighbor w outside this path. Let G be the tree formed from G by deleting the edge w k w k and adding the edge ww k. Obviously, G has diameter. Let V and V be respectively the set of vertices of the subtree of G containing vertex w k and w k formed by deleting the edge w k w k. Let d = dv ij i, v j G and let rc ij be the i, j-entry of the reciprocal complementary matrix of G. For v i, v j V or v i, v j V with i j, we have d ij = d ij and then rc ij < rc ij. For v i V and v j V, we have d ij = d ij or d ij = d ij and then rc ij = rc ij or rc ij < rc ij. Thus, RCWG < RCWG. Using this transformation, we can finally obtain RCWG > RCWP n = n if G P n. By combining Corollary 3, we know that RCW number satisfies the basic requirement to be a branching index []. 3. Nordhaus Gaddum-type result for the reciprocal complementary Wiener number Zhang and Wu [] and Zhou, Cai and Trinajstić [5] obtained the Nordhaus Gaddum-type result for the Wiener index, Zagreb indices, connectivity index and the Harary index, respectively. In the following, we give the Nordhaus Gaddumtype result for RCW number. There is only one connected graph P on vertices with the connected complement P = P. Obviously, RCWP RCWP = RCWP = 6. For n 5, the diameter of P n is. Lemma 8. Let G be a connected graph on n 5 vertices. If G has diameter, then RCWG RCWG n 5n 6 with equality if and only if G = P n. Proof. By Proposition, RCWG n with equality if and only if G = P n. Let [ m be the number ] of edges in G. Then m nn n. By Proposition, RCWG = nn m nn nn n = nn n with equality if and only if the number of edges of G is equal to n. Thus the result follows easily. Lemma 9. Let G be a connected graph on n 5 vertices. Suppose that both G and G have diameter 3. Then RCWGRCWG 5nn with equality if and only if dg, 3 = dg, 3 =. Proof. Let t k = dg, k and t k = dg, k. Obviously, t t 3 = t, t t 3 = t and t t = nn. Then RCWG RCWG = 3 k= t k t k k = t t t t 3 t t 3 t 3 t 3 3 = t t 3 t t 5nn = t 3 t 3 5nn with equality if and only if t 3 = t 3 =. t 3 t 3 = 5 6 t t t 3 t 3 It is easily seen that there are pairs of graphs on n vertices such that both of them have diameter three and t 3 = t 3 =. For example, if n = 5, then there is exactly one pair G and G : the graph formed from the path P 5 by adding an edge between the two neighbors of its center and its complement which is isomorphic to itself such that RCWGRCWG = 8 = 5nn. 3 Proposition 0. Let G be a connected graph on n 5 vertices with a connected G. Then 3nn RCWG RCWG with equality if and only if both G and G have diameter, whilst n 5n 6 for n 9 RCWG RCWG 5nn for 5 n 8 with equality if and only if G = P n or G = P n for n 9, and both G and G have diameter three with dg, 3 = dg, 3 = for 5 n 8.
B. Zhou et al. / iscrete Applied Mathematics 57 009 68 633 633 Proof. Let m and m be respectively the number of edges of G and G. Then m m = nn. By Proposition, nn RCWG RCWG m nn m = nn m m 3nn = with equality if and only if both G and G have diameter. On the other hand, note that either both G and G have diameter 3 or one of them has diameter, and that 5nn > n 5n 6 if and only if n 9. The second part of the proposition follows from Lemmas 8 and 9. Acknowledgements BZ and XC were supported by the National Natural Science Foundation of China Grant No. 067076 and NT by the Ministry of Science, Education and Sports of Croatia Grant No. 098-77095-99. References []. Bonchev, Information Theoretic Indices for Characterization of Chemical Structures, Research Studies Press/Wiley, Chichester, 983, 7 7. []. Bonchev, N. Trinajstić, Information theory, distance matrix, and molecular branching, J. Chem. Phys. 67 977 57 533. [3] I. Gutman, K.C. as, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 00 83 9. [] I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. III. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 7 97 535 538. [5] P.J. Hansen, J.P. Jurs, Chemical applications of graph theory. Part I. Fundamentals and topological indices, J. Chem. Educ. 65 988 57 580. [6] H. Hosoya, Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons, Bull. Chem. Soc. Japan 97 33 339. [7] O. Ivanciuc, QSAR comparative study of Wiener descriptors for weighted molecular graphs, J. Chem. Inf. Comput. Sci. 0 000. [8] O. Ivanciuc, T. Ivanciuc, A.T. Balaban, The complementary distance matrix, a new molecular graph metric, ACH-Models Chem. 37 000 57 8. [9] O. Ivanciuc, T. Ivanciuc, A.T. Balaban, Quantitative structure-property relationship evaluation of structural descriptors derived from the distance and reverse Wiener matrices, Internet Electron. J. Mol. es. 00 67 87. [0] O. Ivanciuc, T. Ivanciuc, A.T. Balaban, Vertex- and edge-weighted molecular graphs and derived structural descriptors, in: J. evillers, A.T. Balaban Eds., Topological Indices and Related escriptors in QSAR and QSPR, Gordon and Breach, The Netherlands, 999, pp. 69 0. []. Janežič, A. Miličević, S. Nikolić, N. Trinajstić, Graph Theoretical Matrices in Chemistry, in: Mathematical Chemistry Monographs, vol. 3, University of Kragujevac, Kragujevac, 007, a pp. 6 6, b pp. 79 80, c pp. 80 8. [] S. Nikolić, G. Kovačević, A. Miličević, N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta 76 003 3. [3] S. Nikolić, N. Trinajstić, Z. Mihalić, The Wiener index: evelopment and applications, Croat. Chem. Acta 68 995 05 8. [] E.A. Nordhaus, J.W. Gaddum, On complementary graphs, Amer. Math. Monthly 63 956 75 77. [5] O. Ore, iameters in graphs, J. Combin. Theory 5 968 75 8. [6] J. Plesník, On the sum of all distance in a graph or digraph, J. Graph Theory 8 96. [7] O.E. Polansky, On the modelling of Wiener numbers, in: A. Graovac Ed., MATH/CHEM/COMP 988, Elsevier, Amsterdam, 989, pp. 67 8. [8].H. Rouvray, Predicting chemistry from topology, Sci. Amer. 55 986 0 7. [9].H. Rouvray, The rich legacy of half a century of the Wiener index, in:.h. Rouvray and R.B. King Eds., Topology in Chemistry iscrete Mathematics of Molecules, Horwood, Chichester, 00, pp. 6 37. [0] R. Todeschini, V. Consonni, Handbook of Molecular escriptors, Wiley-VCH, Weinheim, 000. [] N. Trinajstić, Chemical Graph Theory, nd revised ed., CRC Press, Boca Raton, 99, a pp., b pp. 5 6. [] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 97 7 0. [3] R.J. Wilson, Introduction to Graph Theory, Oliver and Boyd, Edinburgh, 97, a p. 9, b p. 3, c p. 0. [] L. Zhang, B. Wu, The Nordhaus Gaddum-type inequalities for some chemical indices, MATCH Commun. Math. Comput. Chem. 5 005 89 9. [5] B. Zhou, X. Cai, N. Trinajstić, On Harary index, J. Math. Chem. 008 6 68. [6] B. Zhou, I. Gutman, Relationships between Wiener, hyper-wiener and Zagreb indices, Chem. Phys. Lett. 39 00 93 95.