On the Randić index and Diameter of Chemical Graphs

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1 On the Rić index Diameter of Chemical Graphs Dragan Stevanović PMF, University of Niš, Niš, Serbia PINT, University of Primorska, Koper, Slovenia (Received January 4, 008) Abstract Using the AutoGraphiX system, Aouchiche, Hansen Zheng [] proposed a conjecture that the difference the ratio of the Rić index the diameter of a graph are minimum for paths. We prove this conjecture for chemical graphs. Introduction All graphs G = (V, E) will be finite, simple undirected, we follow the stard graph-theoretic terminology, which may be found, for example, in [9]. The degree the neighborhood of a vertex u V will be denoted by d u N u, respectively. The minimum the maximum degree of a graph G are denoted by δ G G, respectively. The diameter the radius of G will be denoted by D G r G, respectively. A chemical graph is a graph with 4. The Rić index Ra(G), also called the connectivity index, of a graph G = (V, E) was introduced by the chemist Milan Rić in 975 [8] as Ra G = uv E. While it was designed to measure the extent of branching of the carbon-atom skeleton of saturated hydrocarbons, it was demonstrated that the Rić index is well correlated dragance06@yahoo.com Supported by the research grant 4405G of the Serbian Ministry of Science Environmental Protection the research program P-085 of the Slovenian Agency for Research.

2 with a variety of physico-chemical properties of alkanes, such as boiling point, enthalpy of formation, surface areas solubility in water. For a comprehensive survey of the (mathematical) properties of Rić index, see the book [4]. A number of mathematical papers devoted to the Rić index deals with finding the extremal graphs having the minimum or the maximum Rić index, for example, [3, 4, 5, 9, 0,,, 3, 6, 7, 0,, ]. On the other h, Aouchiche, Hansen Zheng have studied in [, ] relations between the Rić index other graph invariants using the AutoGraphiX system [6]. One of the conjectures obtained is Conjecture ([]) For any connected graph on n 3 vertices with Rić index Ra diameter D, Ra D n + Ra D n 3 + = n + n, with equalities if only if G is the path P n. This conjecture is somewhat related to Conjecture of Graffiti [7, 8], its modified form proposed in [6] that for all connected graphs G except even paths Ra G r G. In particular, it was proven in [6] that for all trees T, Ra T r T 3, which may be used to approach (although not prove it) the second part of the above conjecture, since in trees either D T = r T or D T = r T. Still, Conjecture may be proved for trees using the result of Li Zhao [6] who described the trees attaining the minimum Rić index when the diameter of a tree is fixed. Theorem ([6]) Among all trees of order n with diameter D, (i) for D 3, Ra T D (n D) (n D + ) +, the equality holds if only if T is a comet CS(n, n D + ); (ii) for D 4 T = CS(n, n D + ), Ra T D 4 + (n D) (n D + ) +, the equality holds if only if T is a path of length D with n D pendent vertices attached to the same vertex of a path. The proof of Conjecture for trees is now a straightforward application of stard calculus techniques to identify the minimum of functions Ra T D T Ra T /D T when D T ranges from to n. Our main goal is to prove the Conjecture for chemical graphs. The starting step of our approach is the fact that a graph G with diameter D has to contain the path P D+ as an induced subgraph. Thus, we will be interested to study under what conditions the induced subgraph of G has smaller Rić index than G? This does not hold in general, not even for all trees. Consider, for example, a tree T in Fig. its vertex v: we have ( 6 = Ra T v > Ra T = 6 + )

3 Figure : The tree whose induced subgraph has larger Rić index. However, as it will turn out, the Rić index of a proper induced subgraph of a chemical graph G is always less than Ra G. This result will be the foundation of the proof of Conjecture for chemical graphs. Rić index of a vertex-deleted subgraph The main topic of this section is the following useful Lemma 3 If 4δ G G in a connected graph G = (V, E), then for any vertex u V Ra G u Ra G. The equality holds if only if u is adjacent to all vertices of G u, δ G = (n )/4, each component of G u is a (δ G )-regular graph. Proof. Let δ = δ G = G. Fix the vertex u V let M u = V \ ({u} N u ) be the set of vertices that are not adjacent to u. Further, let N u denote the set of those vertices of N u which have degree greater than. Then Ra G = + v N du u d v while Ra G u = Therefore, v,w N u,vw E v,w N u,vw E +,w M u,vw E + (d v )(d w ) + +,w M u,vw E (d v )d w Ra G Ra G u = N u N u du + v N u v M u,w M u,vw E, v M u,w M u,vw E ().

4 + + v,w N u,vw E v N u,w M u,vw E (d v )(d w ). (d v )d w Let d v,nu d v,mu be the numbers of neighbors of v within N u M u, respectively. Obviously, d v,nu +d v,mu = d v, as the remaining neighbor u does not belong to N u M u. First, we have (d v )d dv w d w,w M u,vw E = ( ( v N u = < v N u dv ) dv w M u,vw E dw dv ) dv,m u (since d w δ) dv δ ( + ) dv d v dv,m u δ dv d v,mu d v d v,mu d v (d v ) dv,m u δ (as + x < x/) () dv (from δ ) (from d u ). Next, we get = = v,w N u,vw E v,w N u,vw E v,w N u,vw E (d v )(d w ) ( + d v + d w (d v )(d w ) v,w N u,vw E = v N u (d v ) d v + d w (d v )(d w ) ( ) (d v ) + (d w ) w N u,vw E (d v ) dv,n u dv δ d v,nu d v ) (3) (since d w δ) (4) dv (from δ ) (5)

5 v N u d v,nu d v (as d u ). (6) The first inequality + dv+dw dv+dw (d v )(d w ) (d v )(d w ) above is implied by the following chain of inequalities for positive x, y by setting x := d v y := d w : (x + y) 4x y (x + )(y + ) + x + y + + x + y + x + y. Now from () d v,nu + d v,mu = d v we get ( (x + y) = + x + y ) 4x y Ra G Ra G u N u N u + d v,mu du d v d v,nu d v 0. (7) Concerning the case of equality, note first that the equality in (7) may not hold if N u N u or if d v,mu 0 for at least one v N u, due to a strict inequality in (). Thus, as G is connected, a necessary condition for the equality in (7) is that M u =, i.e., that u is adjacent to all vertices of G u that δ G. In such case, equality holds in (7) if only if equality holds in each of (3) (6), which in turn holds if only if each component of G u is (δ )-regular graph for δ = n. 4 Unfortunately, Lemma 3 cannot be generalized to arbitrary induced subgraphs of a graph G with 4δ G G. If we remove an arbitrary vertex u from G, then certainly G G u, but we cannot say anything about δ G u, which may prevent further application of Lemma 3. Namely, it is easy to construct examples of graphs G such that δ G u is either larger than, equal to or smaller than δ G. The situation becomes clearer if we work with chemical graphs: then it always holds that 4δ G 4 G, as we may freely ignore possible isolated vertices of G, since they have no influence on the Rić index whatsoever, take δ G to represent the minimum degree of nonisolated vertices. Further, notice that the connectedness in Lemma 3 is essentially used only to get a more elegant characterization of the case of equality. Thus, from Lemma 3 we easily obtain the following Lemma 4 If G is a chemical graph H its proper induced subgraph, then Ra H < Ra G. 3 The proof of Conjecture for chemical graphs Theorem 5 For any connected chemical graph G on n 3 vertices Ra G D G n + with equalities if only if G = P n. Ra G D G + n,

6 Proof. A connected graph G with diameter D G contains the path P DG + as its induced subgraph [9]. If P DG + = G, then D G = n the equality holds in both inequalities above. If P DG + is a proper induced subgraph of G, then from Lemma 4 Then from D G < n Ra G > Ra PDG + = D G +. Ra G D G > D G + > n + Ra G > D G/ + = D G D G + > D G + n. References [] M. Aouchiche, P. Hansen, M. Zheng, Variable Neighborhood Search for Extremal Graphs. 8. Conjectures Results about the Rić Index, MATCH Commun. Math. Comput. Chem. 56 (006), [] M. Aouchiche, P. Hansen, M. Zheng, Variable Neighborhood Search for Extremal Graphs. 9. Further Conjectures Results about the Rić Index, MATCH Commun. Math. Comput. Chem. 58 (007), [3] O. Araujo, J.A. de la Peña, The connectivity index of a weighted graph, Linear Algebra Appl. 83 (998), [4] B. Bollobás, P. Erdös, Graphs of extremal weights, Ars Combin. 50 (998), [5] G. Caporossi, I. Gutman, P. Hansen, L. Pavlović, Graphs with maximum connectivity index, Comput. Biol. Chem. 7 (003), [6] G. Caporossi, P. Hansen, Variable neighborhood search for extremal graphs.. The AutoGraphiX system, Discrete Math. (000), [7] S. Fajtlowicz, On conjectures of Graffiti, Discrete Math. 7 (988), 3 8. [8] S. Fajtlowicz, Written on the Wall, Version , regularly updated file accessible via from siemion@math.uh.edu. [9] J. Gao, M. Lu, On the Rić index of unicyclic graphs, MATCH Commun. Math. Comput. Chem. 53 (005), [0] I. Gutman, O. Araujo, D.A. Morales, Estimating the connectivity index of a saturated hydrocarbon, Indian J. Chem. 39 (000), [] I. Gutman, O. Miljković, Molecules with smallest connectivity indices, MATCH Commun. Math. Comput. Chem. 4 (000),

7 [] I. Gutman, O. Miljković, G. Caporossi, P. Hansen, Alkanes with small large Rić connectivity indices, Chem. Phys. Lett. 306 (999), [3] P. Hansen, H. Mélot, Variable neighborhood search for extremal graphs. 6. Analyzing bounds for the connectivity index, J. Chem. Inf. Comput. Sci. 43 (003), -4. [4] X. Li, I. Gutman, Mathematical Aspects of Rić-Type Molecular Structure Descriptors, Mathematical Chemistry Monographs, No., Faculty of Science Mathematics, University of Kragujevac, Kragujevac, 006. [5] X. Li, Y. Shi, A Survey on the Rić Index, MATCH Commun. Math. Comput. Chem. 59 (008), [6] X. Li, H. Zhao, Trees with small Rić connectivity indices, MATCH Commun. Math. Comput. Chem. 5 (004), [7] L. Pavlović, I. Gutman, Graphs with extremal connectivity index, Novi Sad J. Math. 3 (00), [8] M. Rić, On characterization of molecular branching, J. Amer. Chem. Soc. 97 (975), [9] D.B. West, Introduction to Graph Theory, Second Edition, Prentice Hall, Upper Saddle River, NJ, USA, 00. [0] X. Wu, L. Zhang, The third minimal Rić index tree with k pendant vertices, MATCH Commun. Math. Comput. Chem. 58 (007), 3. [] P. Yu, An upper bound on the Rić index of trees, J. Math. Study 3 (998), 5 30 (in Chinese). [] L. Zhang, M. Lu, F. Tian, Maximum Rić index on trees with k pendent vertices, J. Math. Chem. 4 (007), 6 7.