Vertex-Weighted Wiener Polynomials for Composite Graphs

Size: px
Start display at page:

Download "Vertex-Weighted Wiener Polynomials for Composite Graphs"

Transcription

1 Also available at ARS MATHEMATICA CONTEMPORANEA 1 (008) Vertex-Weighted Wiener Polynomials for Composite Graphs Tomislav Došlić Faculty of Civil Engineering, University of Zagreb, Kačićeva 6, Zagreb, Croatia Received 7 July 007, accepted February 008, published online 1 July 008 Abstract Recently introduced vertex-weighted Wiener polynomials are a generalization of both vertex-weighted Wiener numbers and ordinary Wiener polynomials. We present here explicit formulae for vertex-weighted Wiener polynomials of the most frequently encountered classes of composite graphs. Keywords: Wiener index, Wiener polynomial, Wiener number, composite graph, graph product. Math. Subj. Class.: 05C1, 05C05, 05C90 1 Introduction The Wiener number (or Wiener index), first introduced and studied by H. Wiener in 197, 3, is one of the first, and most important topological indices, i.e. graphic invariants used in the study of structure-property correlations. It has received lots of attention in chemical 17, 1 and also in mathematical literature 15, 3, 7 5,. Moreover, a fair number of its generalizations and extensions have been introduced and studied, such as the Balaban index 1, the hyper-wiener index of Randić 16, 1, and Schultz s so-called molecular topological index 19. Many of these generalizations involved suitably chosen vertex-weightings. On the other hand, the fact that the Wiener number can be viewed as the unnormalized first moment of the set of shortest-path distances of a graph, motivated the introduction of higher moments 7, 11, 1, and of the so-called Wiener polynomial (8 and independently 18). Sometimes the Wiener polynomial has been called the Hosoya polynomial, as in references 5i and 0. Yet a further extension of the study of Wiener-like graph invariants resulted recently in a unifying approach, achieved by introducing vertex-weighted Wiener polynomials 10. address: doslic@math.hr (Tomislav Došlić) Copyright c 008 DMFA založništvo

2 T. Došlić: Vertex-Weighted Wiener Polynomials for Composite Graphs 67 The fact that many interesting graphs are composed of simpler graphs, that serve as their basic building blocks, prompted interest in the type of relationship between the Wiener number of a composite graph and Wiener numbers of its building blocks 6. This development was followed a couple of years later by an article 0 that established corresponding relationships for Wiener polynomials. The main purpose of the present paper is to bring together in a unifying context the line of research of 10 with that of 6 and 0, by giving in an explicit form the formulae for vertex-weighted Wiener polynomials of the most important classes of composite graphs. The rest of the paper is organized as follows. In section we give the relevant definitions and some preliminary results. In section 3 we state and prove our main results. Finally, those results are illustrated by a few characteristic examples in section. Vertex-Weighted Wiener Polynomials All graphs considered in this paper are simple and connected, unless stated otherwise. For a given graph G we denote by V (G) its vertex set, and by E(G) its edge set. The cardinalities of these two sets are denoted by n and e, respectively, or, if pertaining to a graph G i, by n i and e i, for the corresponding subscript i. The edge e E(G) with the endpoints u and v we denote by (u, v). The shortest-path distance between vertices u and v in a graph G is denoted by D G (u, v), or simply by D(u, v) when there is no possibility of confusion. The degree in G of a vertex u V (G) is denoted by d G (u), or simply by d(u). The (unweighted) Wiener polynomial of G is defined as P o (G; x) = x D(u,v) with x a dummy variable. This coincides with the definition of Hosoya 8, and Sagan et al. 18. A corresponding (singly) vertex-weighted Wiener polynomial is defined P v (G; x) = 1 d(u) + d(v)x D(u,v)+1 and a doubly vertex-weighted Wiener polynomial is P vv (G; x) = d(u)d(v)x D(u,v)+ where d(u) is the degree of vertex u. The Wiener number of a graph G is given by W (G) = D(u, v). The following relationship between the Wiener number and the Wiener polynomial of G was noted in 8, 18: W (G) = P 0(G; x) x=1. The corresponding generalizations for the singly and doubly vertex-weighted cases were given in 10: 1 1 W v (G) = x P v(g; x) ; W vv (G) = x=1 x P vv(g; x). (1) x=1

3 68 Ars Mathematica Contemporanea 1 (008) Here W v (G) and W vv (G) denote singly and doubly vertex-weighted Wiener number of G, defined by W v (G) = d(u) + d(v) D(u, v), W vv (G) = d(u)d(v)d(u, v), respectively. We refer the reader to the same article for more results on vertex-weighted Wiener polynomials of trees and also of the so-called thorny graphs. For a vertex w V (G) we define the partial Wiener polynomial of G with respect to w (or rooted in w): H 0 (G; w; x) = x D(u,w). u V (G) u w In the same manner, we define singly and doubly vertex-weighted partial Wiener polynomials of G rooted at w: H v (G; w; x) = 1 H vv (G; w; x) = u V (G) u w u V (G) u w d(u) + d(w)x D(u,w)+1 ; d(u)d(w)x D(u,w)+. The unweighted partial Wiener polynomial was introduced in 0 and independently in 9, and the weighted ones we modelled after 10. Before we proceed to introduce the composite graphs, let us make the following observation: Lemma 1. Let G be a simple, connected, d-regular graph. Then: P v (G; x) = dxp 0 (G; x); P vv (G; x) = d x P 0 (G; x). The proof is obvious from the definitions of P 0, P v and P vv. Now we introduce the four standard types of composite graphs that we consider in this paper. We define the Cartesian product G 1 G as the graph with the vertex set V (G 1 ) V (G ), with vertices u = (u 1, u ) and v = (v 1, v ) connected by an edge if and only if u 1 = v 1 and (u, v ) E(G ) or u = v and (u 1, v 1 ) E(G 1 ). Lemma. Let G 1 and G be simple, connected graphs. Then D G1 G (u, v) = D G1 (u 1, v 1 ) + D G (u, v ). The proof is straight-forward but somewhat technical, and we refer the reader to 6, where it is worked out in full detail. The Cartesian product of G 1 and G is in mathematical literature usually denoted by G 1 G, but we follow here the notation of Yeh and Gutman 6.

4 T. Došlić: Vertex-Weighted Wiener Polynomials for Composite Graphs 69 The sum of two graphs G 1 and G is defined as a graph G 1 + G with the vertex set V (G 1 + G ) = V (G 1 ) V (G ), and the edge set E(G 1 + G ) = E(G 1 ) E(G ) {(u 1, u ); u 1 V (G 1 ), u V (G )}. In other words, we join every vertex from G 1 to every vertex in G. The sum of two graphs is sometimes also called join, and denoted by G 1 G (0). Obviously, any two vertices of G 1 + G are either at the distance 1 or at the distance. The next binary operation we consider is the composition of two graphs. For given graphs G 1 and G, their composition is the graph G 1 G with the vertex set V (G 1 G ) = V (G 1 ) V (G ), and the vertices u = (u 1, u ) and v = (v 1, v ) of G 1 G are adjacent if and only if u 1 = v 1 G 1 and (u, v ) E(G ), or (u 1, v 1 ) E(G 1 ). In other words, G 1 G is obtained by expanding each vertex of G 1 into a copy of G, with each edge of G 1 replaced by the edge set of the corresponding G + G. The (shortest-path) distance of two vertices u = (u 1, u ) and v = (v 1, v ) of G 1 G is given by D G1 (u 1, v 1 ) u 1 v 1 ; D G1G (u, v) = 1 u 1 = v 1, (u, v ) E(G ); u 1 = v 1, (u, v ) E(G ). Finally, for given graphs G 1 and G we define the cluster G 1 {G } as the graph obtained by taking one copy of G 1 and V (G 1 ) copies of a rooted graph G, and by identifying the root of the i-th copy of G with the i-th vertex of G 1, for i = 1,..., V (G 1 ). We denote the root vertex of G by w, and the copy of G whose root is identified with the vertex u V (G 1 ) by G u. The distance between two vertices y, z G 1 {G } is given by D G (y, z) y G u, z G u ; D G1{G }(y, z) = D G1 (u, v) y = w G u, z = w G v ; D G (y, w) + D G1 (u, v) + D G (w, z) y G u, z G v, u v. There are many more binary operations on graphs, but we restrict our attention to the above four, since their combination and iteration covers most cases of practical interest. The last part of this preliminary section is concerned with Zagreb indices, a pair of topological indices denoted by M 1 (G) and M (G) and introduced some 30 years ago 6. They are defined as follows: M 1 (G) = d(u) ; M (G) = u V (G) (u,v) E(G) d(u)d(v). The first Zagreb index can be also expressed as a sum over the edges of G: M 1 (G) = d(u) + d(v). (u,v) E(G) We refer the reader to 13 for the proof of this fact and for more information on Zagreb indices. Obviously, the Zagreb indices can be viewed as the contributions of pairs of adjacent vertices to vertex-weighted Wiener numbers. Curiously enough, it turns out that similar contributions of non-adjacent pairs of vertices must be taken into account when computing

5 70 Ars Mathematica Contemporanea 1 (008) vertex-weighted Wiener polynomials of certain composite graphs. Hence we propose to call such contributions Zagreb coindices. More formally, the first Zagreb coindex of a graph G is defined by M 1 (G) = d(u) + d(v), and the second Zagreb coindex is given by M (G) = (u,v) E(G) (u,v) E(G) d(u)d(v). The Zagreb indices and Zagreb coindices will be helpful in presenting our main results in a more compact form. 3 Explicit formulae for vertex-weighted Wiener polynomials In this section we state and prove our main results, by giving explicit formulae for vertexweighted Wiener polynomials of composite graphs in terms of weighted and unweighted Wiener polynomials and some simple graphic invariants of underlying components. We remind the reader that n i and e i denote the respective numbers of vertices and of edges of graph G i. Theorem 3. Let G 1 and G be two simple connected graphs. P v (G 1 G ; x) = P v (G 1 ; x)p 0 (G ; x) + P v (G ; x)p 0 (G 1 ; x) + n P v (G 1 ; x) + e xp 0 (G 1 ; x) + n 1 P v (G ; x) + e 1 xp 0 (G 1 ; x); P vv (G 1 G ; x) = P vv (G 1 ; x)p 0 (G ; x) + P v (G 1 ; x)p v (G ; x) + P 0 (G 1 ; x)p vv (G ; x) + n P vv (G 1 ; x) + e xp v (G 1 ; x) + M 1 (G )x P 0 (G 1 ; x) + n 1 P vv (G ; x) + e 1 xp v (G ; x) + M 1 (G 1 )x P 0 (G ; x). Proof. It is obvious from the definition of G 1 G that d G1 G (u 1, u ) = d G1 (u 1 ) + d G (u ). Though in this proof we allow the arguments of various distances and degrees to specify the relevant graphs, so that this becomes d(u 1, u ) = d(u 1 ) + d(u ). Then rephrasing the definition of P v (G; x) as 1 u v d G(u) + d G (v)x DG(u,v)+1, we chose G = G 1 G and

6 T. Došlić: Vertex-Weighted Wiener Polynomials for Composite Graphs 71 further make use of Lemma to obtain P v (G 1 G ; x) = 1 d(u 1 ) + d(u ) + d(v 1 ) + d(v ) x D(u1,v1) x D(u,v) x u 1 v 1 u v + 1 d(u 1 ) + d(v 1 ) + d(u ) x D(u1,v1)+1 u 1 v 1 u =v + 1 d(u ) + d(v ) + d(u 1 ) x D(u,v)+1 u v u 1=v 1 = 1 d(u 1 ) + d(v 1 ) x D(u1,v1)+1 x D(u,v) u 1 v 1 u v + x D(u1,v1) d(u ) + d(v ) x D(u,v)+1 u v + 1 d(u 1 ) + d(v 1 ) x D(u1,v1)+1 + d(u ) + d(v ) x u =v u 1 v d(u ) + d(v ) x D(u,v)+1 + d(u 1 ) + d(v 1 ) x u 1=v 1 u v = 1 P 0 (G ; x) d(u 1 ) + d(v 1 ) x D(u1,v1)+1 + P v (G ; x)x D(u1,v1) u 1 v P v (G 1 ; x) + xp 0 (G 1 ; x) d(u ) u =v + 1 P v (G ; x) + xp 0 (G ; x) d(u 1 ) u 1=v 1 = P v (G 1 ; x)p 0 (G ; x) + P v (G ; x)p 0 (G 1 ; x) + n P v (G 1 ; x) + e xp 0 (G 1 ; x) + n 1 P v (G ; x) + e 1 xp 0 (G ; x). u 1 v 1 x D(u1,v1) u v x D(u,v) To achieve the last equality we have used the fact that u V (G) d(u) = e, for a general graph G. And the first result of the theorem is attained. In order to prove our second claim, we start from the rephrased definition of P vv (G; x) u v d G(u)d G (v)x DG(u,v)+ and take G = G 1 G. as 1 P vv (G 1 G ; x) = 1 d(u 1 ) + d(v 1 ) d(u ) + d(v ) x D(u1,v1)+D(u,v)+. u v Again, the sum can be split into three sums, 1, u 1 v 1 u v 1, u 1=v 1 u v and 1, u =v u 1 v 1

7 7 Ars Mathematica Contemporanea 1 (008) which we respectively denote by S 1, S, and S 3. Now we have: S 1 = 1 d(u 1 )d(u )x D(u1,v1)+ u 1 v 1 + d(u 1 )x D(u1,v1)+1 u v x D(u,v) d(v )x D(u,v)+1 u v + d(v 1 )x D(u1,v1)+1 d(u )x D(u,v)+1 u v + x D(u1,v1) d(u )d(v )x D(u,v)+ u v = P vv (G 1 ; x)p 0 (G ; x) + P v (G 1 ; x)p v (G ; x) + P 0 (G 1 ; x)p vv (G ; x), where u v d(u)xd(u,v))+1 = P v (G; x) has been used. Next S = 1 d(u 1 ) x D(u,v) u 1=v 1 u v + 1 d(u 1 )x d(v )x D(u,v)+1 + d(u )x D(u,v)+1 u 1=v 1 u v u v + 1 x D(u1,v1) d(u )d(v )x D(u,v)+ u 1=v 1 u v = x P 0 (G; x) u 1 d(u 1 ) + xp v (G ); x) u 1 d(u 1 ) + P vv (G ; x) u 1 1 = M 1 (G 1 )x P 0 (G ; x) + e 1 xp v (G ; x) + n 1 P vv (G ; x). Now because of the symmetry between S and S 3, one obtains S 3 by switching the roles of G 1 and G in S. Addition of S 1, S, and S 3 yields the second result of the theorem. Theorem. Let G 1 and G be two simple graphs. 1 P v (G 1 + G ; x) = M 1 (G 1 ) + M 1 (G ) ( n1 ) + n e 1 + ( n ) + n 1 e x 3 1 M 1(G 1 ) + M 1 (G ) + (e 1 n + n 1 e ) + 1 n 1n (n 1 + n ) x ; P vv (G 1 + G ; x) = M (G 1 ) + M (G ) + n M 1 (G 1 ) + n 1 M 1 (G ) + n 1 n ( n ) e + ( n1 ) e 1 x + M (G 1 ) + M (G ) + n M 1 (G 1 ) + n 1 M 1 (G )+ n 1n + 3e 1 n + 3e n 1 + e 1 e x 3.

8 T. Došlić: Vertex-Weighted Wiener Polynomials for Composite Graphs 73 Proof. As already observed, all vertices of G 1 + G are either at distance 1, or at distance. The vertices at distance are precisely those of G 1 that are not adjacent in G 1, and those of G that are not adjacent in G. Furthermore, for a vertex u V (G 1 ) we have d G1+G (u) = d G1 (u) + n, and for a vertex v V (G ), we have d G1+G (v) = d G (v) + n 1. Hence, we can conclude that P v (G 1 + G ; x) = a 3 x 3 + a x, and P vv (G 1 + G ; x) = b x + b 3 x 3. The coefficients a, a 3, b 3, and b can be determined by a straightforward computation. For example, a 3 = 1 = 1 = 1 Similarly, a = 1 G 1 (u,v) E(G 1 ) + 1 e E(G 1) d(u) + d(v) + n + 1 d(u) + d(v) + 1 e E(G ) M 1 (G 1 ) + M 1 (G ) + n ( n1 (u,v) E(G 1 ) d(u) + d(v) + n + 1 v G d(u) + d(v) + n 1 + n G (u,v) E(G ) d(u) + d(v) + n 1 d(u) + d(v) + n E(G1 ) + n1 E(G ) ) e 1 = 1 M 1(G 1 ) + n e M 1(G ) + n 1 e + 1 (u,v) E(G ) ( n ) + n 1 e. d(u) + d(v) + n 1 = 1 M 1(G 1 ) + M 1 (G ) + (e 1 n + n 1 e ) + 1 n 1n (n 1 + n ). d(u)n + n + n 1 n + e The coefficients b and b 3 can be determined in a completely analogous manner, though we omit the details. Theorem 5. Let G 1 and G be simple graphs, and let G 1 be connected. Then: P v (G 1 G ; x) = n 3 P v (G 1 ; x) + n e xp 0 (G 1 ; x) n1 + M 1(G ) + e 1 e n x n1 ( + M n ) 1(G ) + e 1 n e x 3 ; P vv (G 1 G ; x) = n P vv (G 1 ; x) + e n xp v (G 1 ; x) + e x P 0 (G 1 ; x) Proof. We start from the observation + n 1 M (G ) + e 1 n M 1 (G ) + e n M 1 (G 1 ) x 3 ( + n 1 M (G ) + e 1 n M 1 (G ) + n n ) e M 1 (G 1 ) x. d G1G (u 1, u ) = d G (u ) + n d G1 (u 1 ).

9 7 Ars Mathematica Contemporanea 1 (008) Taking into account our formula for distance in G 1 G, given immediately after the definition of composition of two graphs, we have the following: P v (G 1 G ; x) = 1 d(u) + d(v) x D(u,v)+1 u v = 1 d(u ) + d(v ) x x D(u1,v1)+1 u v u 1 v 1 + n d(u 1 ) + d(v 1 ) x D(u1,v1)+1 u 1 v d(u ) + d(v ) + n d(u 1 ) + d(v 1 ) 1 u 1=v 1 e E(G ) e E(G ) + 1 d(u ) + d(v ) + n d(u 1 ) + d(v 1 ) 1 u =v e E(G ) e E(G ) = 1 xp 0 (G 1 ; x) d(u ) + d(v ) + n P v (G 1 ; x) 1 u v u v + 1 M 1 (G ) + e n d(u 1 ) + d(v 1 ) x u 1=v M 1 (G ) + n E((G) ) d(u 1 ) + d(v 1 ) x 3 u 1=v 1 u 1=v 1 = n 3 P v (G 1 ; x) + n e xp 0 (G 1 ; x) n1 + M 1(G ) + e 1 e n x n1 ( + M n ) 1(G ) + e 1 n e x 3 ; The expression for P vv (G 1 G ; x) is obtained in much the same manner, and we omit the details. Theorem 6. Let G 1 and G be simple connected graphs. Let the copies of G used in the construction of G 1 {G } be rooted in vertex w, and let d G (w) = t. Then: P v (G 1 {G }; x) = H 0 (G ; w; x) + 1 P v (G 1 ; x) + n 1 P v (G ; x) + H 0 (G ; w; x) H v (G ; w; x) txh 0 (G ; w; x) + H v (G ; w; x) + tx P 0 (G 1 ; x) + e 1 xh 0 (G ; w; x); P vv (G 1 {G }; x) = P vv (G 1 ; x) + n 1 P vv (G ; x) x x 3

10 T. Došlić: Vertex-Weighted Wiener Polynomials for Composite Graphs 75 + H v (G ; w; x) txh 0 (G ; w; x) + tx P v (G 1 ; x) + H vv (G ; w; x) + H v (G ; w; x) txh 0 (G ; w; x) + t x P 0 (G 1 ; x) + e 1 x H v (G ; w; x) txh 0 (G ; w; x). Proof. Here we follow the fivefold path employed in 10. We partition our sum in five sums as follows. The first sum S 1 consists of contributions to P v (G 1 {G }; x) of pairs of vertices from G 1. For a vertex u V (G 1 ), we have d G1{G }(u) = d G1 (u) + t, since u serves as an anchor-point for a copy of G, and the root vertex of this copy has degree t. Taking this into account, we obtain S 1 = 1 G 1 d(u) + d(v) x D(u,v)+1 = 1 G 1 d G1 (u) + d G1 (v) x D(u,v) G 1 tx = P v (G 1 ; x) + txp 0 (G 1 ; x). x D(u,v) The second sum S is taken over all pairs of vertices such that one of them, u, is in G 1, and the other one is in G u, the copy of G that is anchored at u. Thence d G1 (u) + d G1 (v) x D(u,v)+1 S = 1 = 1 v G u v u d G1 (u)x D(u,v) d G (v) + t x D(u,v)+1 v G u v u = e 1 xh 0 (G ; w; x) + n 1 H v (G ; w; x). The third sum S 3 is over all pairs of vertices such that one of them, u, is in G 1, and the other one is in some G z, where z u. S 3 = 1 d(u) + d(v) x D(u,v)+1 v G u v u = 1 v G z z u d G1 (u) + t + d(v) x D(u,z)+D(z,v) z G 1 v G z d G1 (z) + t + d(v) x D(u,z)+D(z,v)+1 z G 1 v G u = 1 H 0(G ; w; x) d G1 (u) + d G1 (z) x D(u,z)+1 z G H v(g ; w; x) x D(u,z) + z G 1 x D(z,u) z G 1

11 76 Ars Mathematica Contemporanea 1 (008) = H 0 (G ; w; x)p v (G 1 ; x) + H v (G ; w; x)p 0 (G 1 ; x). The fourth sum S is taken over all pairs of non-root vertices in G u, and then over all u G 1. S = 1 G v<z v u,z u d G (v) + d G (z) x D(v,z)+1 = 1 P v (G ; x) H v (G ; w; x) = n 1 P v (G ; x) n 1 H v (G ; w; x). Finally, the fifth sum S 5 is over all pairs of non-root vertices from different copies of G. For such a pair v G u, z G y, we have D(v, z) = D G (v, u) + D G1 (u, y) + D G (y, z) = D G (v, w) + D G1 (u, y) + D G (w, z). Now, S 5 = 1 d(v) + d(z) x D(v,z)+1 = 1 = 1 = 1 y G 1 v G u z G y y G 1 v G u z G y d G (v) + d G (z) + t t x D(v,u) x D(u,y) x D(y,z) x y G 1 H 0 (G ; w; x)x D(u,y) txx D(u,y) H 0 (G ; w; x) + H v (G ; w; x)x D(u,y) v G u v G u v G u x D(v,u) x D(v,u) t + d G (v) x D(v,u)+1 H0 (G ; w; x)h v (G ; w; x) txh 0 (G ; w; x) = H 0 (G ; w; x) H v (G ; w; x) txh 0 (G ; w; x) P 0 (G 1 ; x). x D(u,y) y G 1 The formula for P v (G 1 {G }; x) follows upon addition of the quantities S 1, S, S 3, S, and S 5. The formula for P vv (G 1 {G }; x) follows in much the same way. Corrolaries and examples In this section our theorems for vertex-weighted Wiener polynomials are illustrated for several more particular composite graphs. Let P n and C n denote the path and the cycle on n vertices, respectively. It is well known, and it can be easily checked, that their unweighted Wiener polynomials are given by the expressions: n 1 P 0 (P n ; x) = (n k)x k ; k=1

12 T. Došlić: Vertex-Weighted Wiener Polynomials for Composite Graphs 77 n/ 1 P 0 (C n ; x) = n x k + a n x n/, k=1 where a n = n for n even, and a n = n for n odd. Singly and doubly vertex-weighted Wiener polynomials of P n are given by and P v (P n ; x) = (x + 1)P 0 (P n ; x) (n 1)x P vv (P n ; x) = (x + 1) P 0 (P n ; x) (3n )x (n 1)x, respectively. This follows from results given in 10 that give weighted Wiener polynomials of trees in terms of unweighted ones. The expressions for the vertex-weighted Wiener polynomials of the m n rectangular grid follow as this is the Cartesian product graph P m P n. Corollary 7. P v (P m P n ; x) = (x + 1) P 0 (P m ; x)p 0 (P n ; x) + np 0 (P m ; x) + mp 0 (P n ; x) (mn m n)x; P vv (P m P n ; x) = 8(x + 1) P 0 (P m ; x)p 0 (P n ; x) + ( n)x + x + n P 0 (P m ; x) + ( m)x + x + m P 0 (P n ; x) + (m + n 5mn )x (mn m n)x. For m = n, one has the square grid: Corollary 8. P v (P n P n ; x) = (x + 1)P 0 (P n ; x) + n(x + 1)P 0 (P n ; x) n(n 1)x; P vv (P n P n ; x) = 8(x + 1) P 0 (P n ; x) + ( n)x + x + np 0 (P n ; x) (5n 8n + )x n(n 1)x. For m =, one has the ladder graph on n rungs, L n. Corollary 9. P v (L n ; x) = (x + 1)(x + 1)P 0 (P n ; x) (n 1)x; P vv (L n ; x) = (x + 1)(x + 1) P 0 (P n ; x) (n )x 3 (6n 7)x (n 1)x. by It follows from Lemma 1 that the vertex-weighted Wiener polynomials of C n are given P v (C n ; x) = xp 0 (C n ; x); P vv (C n ; x) = x P 0 (C n ; x). Using this fact and Theorem 3, we can compute vertex-weighted Wiener polynomials of a cylinder graph P m C n.

13 78 Ars Mathematica Contemporanea 1 (008) Corollary 10. P v (P m C n ; x) = (3x + 1)P 0 (P m ; x)p 0 (C n ; x) + n(3x + )P 0 (P m ; x) + mxp 0 (C n ; x) n(m 1)x; P vv (P m C n ; x) = (3x + 1) P 0 (P m ; x)p 0 (C n ; x) + n(3x + 1) P 0 (P m ; x) + (m + 1)x m + 1P 0 (C n ; x) + n(8 7m)x n(m 1)x. For m = in Corrolary 10, vertex-weighted Wiener polynomials of an n sided prism, P r n, result: Corollary 11. P v (P r n ; x) = 6x(x + 1)P 0 (C n ; x) + 3nx ; P vv (P r n ; x) = 18x (x + 1)P 0 (C n ; x) + 9nx 3. Corrolary 11 could have also been obtained by first computing P 0 (P r n ; x) via theorem 1 of reference 0, and then using 3-regularity of P r n and applying Lemma 1. The same approach gives us expressions for vertex-weighted Wiener polynomials of a (-regular) torus graph C m C n. Corollary 1. P v (C m C n ; x) = xp 0 (C m ; x)p 0 (C n ; x) + np 0 (C m ; x) + mp 0 (C n ; x); P vv (C m C n ; x) = 16x P 0 (C m ; x)p 0 (C n ; x) + np 0 (C m ; x) + mp 0 (C n ; x). Next, we give vertex-weighted Wiener polynomials for the fence graph P n K. Corollary 13. P v (P n K ; x) = (3x + )P 0 (P n ; x) + (5n )x 8(n 1)x; P vv (P n K ; x) = (3x + ) P 0 (P n ; x) + (5n 3)x 3 16(n 5)x 16(n 1)x. For a closed fence, C n K, we get the following: Corollary 1. P v (C n K ; x) = 0xP 0 (C n ; x) + 5x ; P vv (C n K ; x) = 100x P 0 (C n ; x) + 5x 3. The result can be obtained either from Theorem 5, of by combining Theorem A and Lemma 1. Our last example is concerned with t-fold bristled graph of P n. (For a given graph G, its t-fold bristled graph is obtained by attaching t vertices of degree 1 to each vertex of G.) This graph can be represented as the cluster of P n and the t-star graph K 1,t. Corollary 15. P v (P n {K 1,t }; x) = (x + 1)(tx + 1) P 0 (P n ; x) ( ) ( ) t t n x 3 + n x (n 1)x; ( ) t P vv (P n {K 1,t }; x) = (x + 1) (tx + 1) P 0 (P n ; x) + n x + nt x 3 t(n 1) + 3n x (n 1)x.

14 T. Došlić: Vertex-Weighted Wiener Polynomials for Composite Graphs 79 The result follows by a straightforward application of Theorem 6, taking K 1,t to be rooted in its mid-vertex w, and taking into account the following formulae for its partial Wiener polynomials: ( ) t + 1 H 0 (K 1,t ; w; x) = tx; H v (K 1,t ; w; x) = x ; H vv (K 1,t ; w; x) = t x 3. The t-fold bristled graph of a cycle on n vertices, C n {K 1,t }, is the same object as the (t )-thorny graph of the n-cycle 10, and it can be easily checked that the formulae for P v (C n {K 1,t }; x) and P vv (C n {K 1,t }; x) coincide with those given in 10 for this thorny graph. We conclude by pointing out that the vertex-weighted Wiener numbers of the considered composite graphs can now be easily computed by using formulae (1). For example, from (1) and Corrolary 1 it follows at once that W v (C m C n ) = d dx P 0(C m ; x)p 0 (C n ; x) + np 0 (C m ; x) + mp 0 (C n ; x) x=1 = 8 P 0 (C m ; x)p 0(C n ; x) x=1 + 8 P 0(C m ; x)p 0 (C n ; x) x=1 + nw (C m ) + mw (C n ) ( m ) = 8 W (C n ) + 8 = n W (C m ) + m W (C n ). ( n ) W (C m ) + nw (C m ) + mw (C n ) The Wiener number of the n-cycle is 1 8 n3 for n even, and 1 8 (n3 n) for n odd. Taking both n and m even in the above example yields W v (C m C n ) = 1 m n (m + n). The given theorems might be specialized to yet further special pairs of graphs. 5 Acknowledgments The author is thankful to Prof. D.J. Klein for helpful comments and suggestions and to two anonymous referees for their careful reading and valuable suggestions. The support of the Welch Foundation of Houston, Texas, via Grant # BD-089 is acknowledged. Partial support of the Ministry of Science, Education and Sport of the Republic of Croatia (Grants No and ) is gratefully acknowledged. References 1 A.T. Balaban, Topological indices based on topological distances in molecular graphs, Pure & Appl. Chem. 55 (1983), F. Buckley and F. Harary, Distances in Graphs, Addison-Wesley, Redwood, A.A. Dobrynin, R. Entringer and I. Gutman, Wiener Index of Trees: Theory and Applications, Acta Appl. Math. 66 (001), E. Estrada, Generalization of topological indices, Chem. Phys. Lett. 336 (001), I. Gutman, S. Klavžar, M. Petkovšek and P. Žigert, On Hosoya polynomials of benzenoid graphs, Commun. Math. Chem. (MATCH) 3 (001), 9 66.

15 80 Ars Mathematica Contemporanea 1 (008) I. Gutman and N. Trinajstić, Chem. Phys. Lett. 17 (197), I. Gutman, Y-N. Yeh, Y-N. Lee and Y-L Lau, Some recent results in the theory of the Wiener number, Ind. J. Chem 3 (1993), H. Hosoya, On some counting polynomials in chemistry, Discrete Appl. Math. 19 (1988), O. Ivanciuc, D.J. Klein, Building Block Computation of Wiener-Type Indices for the Virtual Screening of Combinatorial Libraries, Croat. Chem. Acta 75 (00), D.J. Klein, T. Došlić and D. Bonchev, Vertex-weightings for distance-moments and thorny graphs, Discrete Appl. Math. 155 (007), D.J. Klein, I. Gutman, Wiener-Number-Related Sequences, J. Chem. Inf. & Comp. Sci. 39 (1999), D.J. Klein, I. Lukovits, I. Gutman, On the Definition of the Hyper-Wiener Index for Cycle- Containing Structures, J. Chem. Inf. & Comp. Sci. 35 (1995), S. Nikolić, G. Kovačević, A. Miličević and N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta 76 (003), S. Nikolić, N. Trinajstić and Z. Mihalić, The Wiener index: Development and applications, Croat. Chem. Acta 68 (1995), O.E. Polansky and D. Bonchev, The Wiener number of graphs. 1., Commun. Math. Chem. (MATCH) 1 (1986), M. Randić, Novel molecular descriptor for structure-property studies, Chem. Phys. Lett. 11 (1993), D.H. Rouvray, The role of topological distance matrix in chemistry, in Mathematical and Computational Concepts in Chemistry, (ed. N. Trinajstić), Ellis Horwood, Chichester, 1986, B.E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Intl. J. Quantum Chem. 60 (1996), H.P. Schultz, Topological organic chemistry. 1. Graph theory and topological indices for alkanes, J. Chem. Inf. & Comp. Sci. 9 (1989), D. Stevanović, Hosoya polynomials of composite graphs, Discrete Math. 35 (001), S.S. Tratch, M.I. Stankevitch, N.S. Zefirov, Combinatorial Models and Algorithms in Chemistry the Expanded Wiener Number, J. Comp. Chem. 11 (1990), H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (197), H. Wiener, Relations of the physical properties of the isomeric alkanes to molecular structure: surface tension, specific dispersion, and critical solution temperature in aniline, J. Phys. Chem. 5 (198), W. Yan, B-Y. Yang, Y-N. Yeh, The behavior of Wiener indices and polynomials of graphs under five graph decorations, Appl. Math. Lett. 0 (007), B-Y. Yang, Y-N. Yeh, A Crowning Moment for Wiener Indices, Stud. Appl. Math. 11 (00), Y-N. Yeh and I. Gutman, On the sum of all distances in composite graphs, Discrete Math. 135 (199), Y. Zhu, D.J. Klein, I. Lukovits, Extensions of the Wiener Number, J. Chem. Inf. & Comp. Sci. 36 (1996), 0 8.

VERTEX-WEIGHTED WIENER POLYNOMIALS OF SUBDIVISION-RELATED GRAPHS. Mahdieh Azari, Ali Iranmanesh, and Tomislav Došlić

VERTEX-WEIGHTED WIENER POLYNOMIALS OF SUBDIVISION-RELATED GRAPHS. Mahdieh Azari, Ali Iranmanesh, and Tomislav Došlić Opuscula Math. 36, no. (06), 5 3 http://dx.doi.org/0.7494/opmath.06.36..5 Opuscula Mathematica VERTEX-WEIGHTED WIENER POLYNOMIALS OF SUBDIVISION-RELATED GRAPHS Mahdieh Azari, Ali Iranmanesh, and Tomislav

More information

HOSOYA POLYNOMIAL OF THORN TREES, RODS, RINGS, AND STARS

HOSOYA POLYNOMIAL OF THORN TREES, RODS, RINGS, AND STARS Kragujevac J. Sci. 28 (2006) 47 56. HOSOYA POLYNOMIAL OF THORN TREES, RODS, RINGS, AND STARS Hanumappa B. Walikar, a Harishchandra S. Ramane, b Leela Sindagi, a Shailaja S. Shirakol a and Ivan Gutman c

More information

Extremal Graphs with Respect to the Zagreb Coindices

Extremal Graphs with Respect to the Zagreb Coindices MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 65 (011) 85-9 ISSN 0340-653 Extremal Graphs with Respect to the Zagreb Coindices A. R. Ashrafi 1,3,T.Došlić,A.Hamzeh

More information

On a Class of Distance Based Molecular Structure Descriptors

On a Class of Distance Based Molecular Structure Descriptors On a Class of Distance Based Molecular Structure Descriptors Franka Miriam Brückler a, Tomislav Došlić b, Ante Graovac c, Ivan Gutman d, a Department of Mathematics, University of Zagreb, Bijenička 30,

More information

(Received: 19 October 2018; Received in revised form: 28 November 2018; Accepted: 29 November 2018; Available Online: 3 January 2019)

(Received: 19 October 2018; Received in revised form: 28 November 2018; Accepted: 29 November 2018; Available Online: 3 January 2019) Discrete Mathematics Letters www.dmlett.com Discrete Math. Lett. 1 019 1 5 Two upper bounds on the weighted Harary indices Azor Emanuel a, Tomislav Došlić b,, Akbar Ali a a Knowledge Unit of Science, University

More information

Discrete Applied Mathematics

Discrete Applied Mathematics 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

More information

COUNTING RELATIONS FOR GENERAL ZAGREB INDICES

COUNTING RELATIONS FOR GENERAL ZAGREB INDICES Kragujevac Journal of Mathematics Volume 38(1) (2014), Pages 95 103. COUNTING RELATIONS FOR GENERAL ZAGREB INDICES G. BRITTO ANTONY XAVIER 1, E. SURESH 2, AND I. GUTMAN 3 Abstract. The first and second

More information

Computing distance moments on graphs with transitive Djoković-Winkler s relation

Computing distance moments on graphs with transitive Djoković-Winkler s relation omputing distance moments on graphs with transitive Djoković-Winkler s relation Sandi Klavžar Faculty of Mathematics and Physics University of Ljubljana, SI-000 Ljubljana, Slovenia and Faculty of Natural

More information

Counting the average size of Markov graphs

Counting the average size of Markov graphs JOURNAL OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4866, ISSN (o) 2303-4947 www.imvibl.org /JOURNALS / JOURNAL Vol. 7(207), -6 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA

More information

Topological indices: the modified Schultz index

Topological indices: the modified Schultz index Topological indices: the modified Schultz index Paula Rama (1) Joint work with Paula Carvalho (1) (1) CIDMA - DMat, Universidade de Aveiro The 3rd Combinatorics Day - Lisboa, March 2, 2013 Outline Introduction

More information

Upper Bounds for the Modified Second Multiplicative Zagreb Index of Graph Operations Bommanahal Basavanagoud 1, a *, Shreekant Patil 2, b

Upper Bounds for the Modified Second Multiplicative Zagreb Index of Graph Operations Bommanahal Basavanagoud 1, a *, Shreekant Patil 2, b Bulletin of Mathematical Sciences and Applications Submitted: 06-03- ISSN: 78-9634, Vol. 7, pp 0-6 Revised: 06-08-7 doi:0.805/www.scipress.com/bmsa.7.0 Accepted: 06-08-6 06 SciPress Ltd., Switzerland Online:

More information

THE EDGE VERSIONS OF THE WIENER INDEX

THE EDGE VERSIONS OF THE WIENER INDEX MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 61 (2009) 663-672 ISSN 0340-6253 THE EDGE VERSIONS OF THE WIENER INDEX Ali Iranmanesh, a Ivan Gutman, b

More information

A new version of Zagreb indices. Modjtaba Ghorbani, Mohammad A. Hosseinzadeh. Abstract

A new version of Zagreb indices. Modjtaba Ghorbani, Mohammad A. Hosseinzadeh. Abstract Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Filomat 26:1 2012, 93 100 DOI: 10.2298/FIL1201093G A new version of Zagreb indices Modjtaba

More information

Some results on the reciprocal sum-degree distance of graphs

Some results on the reciprocal sum-degree distance of graphs J Comb Optim (015) 30:435 446 DOI 10.1007/s10878-013-9645-5 Some results on the reciprocal sum-degree distance of graphs Guifu Su Liming Xiong Xiaofeng Su Xianglian Chen Published online: 17 July 013 The

More information

Zagreb indices of block-edge transformation graphs and their complements

Zagreb indices of block-edge transformation graphs and their complements Indonesian Journal of Combinatorics 1 () (017), 64 77 Zagreb indices of block-edge transformation graphs and their complements Bommanahal Basavanagoud a, Shreekant Patil a a Department of Mathematics,

More information

Wiener Index of Graphs and Their Line Graphs

Wiener Index of Graphs and Their Line Graphs JORC (2013) 1:393 403 DOI 10.1007/s40305-013-0027-6 Wiener Index of Graphs and Their Line Graphs Xiaohai Su Ligong Wang Yun Gao Received: 9 May 2013 / Revised: 26 August 2013 / Accepted: 27 August 2013

More information

Enumeration of subtrees of trees

Enumeration of subtrees of trees Enumeration of subtrees of trees Weigen Yan a,b 1 and Yeong-Nan Yeh b a School of Sciences, Jimei University, Xiamen 36101, China b Institute of Mathematics, Academia Sinica, Taipei 1159. Taiwan. Theoretical

More information

Wiener Indices and Polynomials of Five Graph Operators

Wiener Indices and Polynomials of Five Graph Operators Wiener Indices and Polynomials of Five Graph Operators Weigen Yan School of Sciences, Jimei University Xiamen 36101, China, and Academia Sinica Taipei, Taiwan wgyan@math.sinica.edu.tw Yeong-Nan Yeh Inst.

More information

SOME VERTEX-DEGREE-BASED TOPOLOGICAL INDICES UNDER EDGE CORONA PRODUCT

SOME VERTEX-DEGREE-BASED TOPOLOGICAL INDICES UNDER EDGE CORONA PRODUCT ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N. 38 07 8 9 8 SOME VERTEX-DEGREE-BASED TOPOLOGICAL INDICES UNDER EDGE CORONA PRODUCT I. Rezaee Abdolhosseinzadeh F. Rahbarnia M. Tavaoli Department of Applied

More information

Iranian Journal of Mathematical Chemistry, Vol. 4, No.1, March 2013, pp On terminal Wiener indices of kenograms and plerograms

Iranian Journal of Mathematical Chemistry, Vol. 4, No.1, March 2013, pp On terminal Wiener indices of kenograms and plerograms Iranian Journal of Mathematical Chemistry, Vol. 4, No.1, March 013, pp. 77 89 IJMC On terminal Wiener indices of kenograms and plerograms I. GUTMAN a,, B. FURTULA a, J. TOŠOVIĆ a, M. ESSALIH b AND M. EL

More information

Hexagonal Chains with Segments of Equal Lengths Having Distinct Sizes and the Same Wiener Index

Hexagonal Chains with Segments of Equal Lengths Having Distinct Sizes and the Same Wiener Index MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 78 (2017) 121-132 ISSN 0340-6253 Hexagonal Chains with Segments of Equal Lengths Having Distinct Sizes and

More information

THE HARMONIC INDEX OF EDGE-SEMITOTAL GRAPHS, TOTAL GRAPHS AND RELATED SUMS

THE HARMONIC INDEX OF EDGE-SEMITOTAL GRAPHS, TOTAL GRAPHS AND RELATED SUMS Kragujevac Journal of Mathematics Volume 08, Pages 7 8. THE HARMONIC INDEX OF EDGE-SEMITOTAL GRAPHS, TOTAL GRAPHS AND RELATED SUMS B. N. ONAGH Abstract. For a connected graph G, there are several related

More information

Extremal Topological Indices for Graphs of Given Connectivity

Extremal Topological Indices for Graphs of Given Connectivity Filomat 29:7 (2015), 1639 1643 DOI 10.2298/FIL1507639T Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Extremal Topological Indices

More information

On the difference between the revised Szeged index and the Wiener index

On the difference between the revised Szeged index and the Wiener index On the difference between the revised Szeged index and the Wiener index Sandi Klavžar a,b,c M J Nadjafi-Arani d June 3, 01 a Faculty of Mathematics and Physics, University of Ljubljana, Slovenia sandiklavzar@fmfuni-ljsi

More information

On the Leap Zagreb Indices of Generalized xyz-point-line Transformation Graphs T xyz (G) when z = 1

On the Leap Zagreb Indices of Generalized xyz-point-line Transformation Graphs T xyz (G) when z = 1 International J.Math. Combin. Vol.(018), 44-66 On the Leap Zagreb Indices of Generalized xyz-point-line Transformation Graphs T xyz (G) when z 1 B. Basavanagoud and Chitra E. (Department of Mathematics,

More information

M-Polynomial and Degree-Based Topological Indices

M-Polynomial and Degree-Based Topological Indices M-Polynomial and Degree-Based Topological Indices Emeric Deutsch Polytechnic Institute of New York University, United States e-mail: emericdeutsch@msn.com Sandi Klavžar Faculty of Mathematics and Physics,

More information

Extremal Zagreb Indices of Graphs with a Given Number of Cut Edges

Extremal Zagreb Indices of Graphs with a Given Number of Cut Edges Graphs and Combinatorics (2014) 30:109 118 DOI 10.1007/s00373-012-1258-8 ORIGINAL PAPER Extremal Zagreb Indices of Graphs with a Given Number of Cut Edges Shubo Chen Weijun Liu Received: 13 August 2009

More information

CCO Commun. Comb. Optim.

CCO Commun. Comb. Optim. Communications in Combinatorics and Optimization Vol. 1 No. 2, 2016 pp.137-148 DOI: 10.22049/CCO.2016.13574 CCO Commun. Comb. Optim. On trees and the multiplicative sum Zagreb index Mehdi Eliasi and Ali

More information

TWO TYPES OF CONNECTIVITY INDICES OF THE LINEAR PARALLELOGRAM BENZENOID

TWO TYPES OF CONNECTIVITY INDICES OF THE LINEAR PARALLELOGRAM BENZENOID NEW FRONT. CHEM. (04) Former: Ann. West Univ. Timisoara Series Chem. Volume 3, Number, pp. 73-77 ISSN: 4-953 ISSN 393-7; ISSN-L 393-7 West University of Timișoara Article TWO TYPES OF CONNECTIVITY INDICES

More information

Average distance, radius and remoteness of a graph

Average distance, radius and remoteness of a graph Also available at http://amc-journal.eu ISSN 855-3966 (printed edn.), ISSN 855-397 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (0) 5 Average distance, radius and remoteness of a graph Baoyindureng

More information

More on Zagreb Coindices of Composite Graphs

More on Zagreb Coindices of Composite Graphs International Mathematical Forum, Vol. 7, 2012, no. 14, 669-673 More on Zagreb Coindices of Composite Graphs Maolin Wang and Hongbo Hua Faculty of Mathematics and Physics Huaiyin Institute of Technology

More information

A Survey on Comparing Zagreb Indices

A Survey on Comparing Zagreb Indices MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 65 (2011) 581-593 ISSN 0340-6253 A Survey on Comparing Zagreb Indices Bolian Liu and Zhifu You School of

More information

RELATION BETWEEN WIENER INDEX AND SPECTRAL RADIUS

RELATION BETWEEN WIENER INDEX AND SPECTRAL RADIUS Kragujevac J. Sci. 30 (2008) 57-64. UDC 541.61 57 RELATION BETWEEN WIENER INDEX AND SPECTRAL RADIUS Slavko Radenković and Ivan Gutman Faculty of Science, P. O. Box 60, 34000 Kragujevac, Republic of Serbia

More information

On Trees Having the Same Wiener Index as Their Quadratic Line Graph

On Trees Having the Same Wiener Index as Their Quadratic Line Graph MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 76 (2016) 731-744 ISSN 0340-6253 On Trees Having the Same Wiener Index as Their Quadratic Line Graph Mohammad

More information

Extremal Kirchhoff index of a class of unicyclic graph

Extremal Kirchhoff index of a class of unicyclic graph South Asian Journal of Mathematics 01, Vol ( 5): 07 1 wwwsajm-onlinecom ISSN 51-15 RESEARCH ARTICLE Extremal Kirchhoff index of a class of unicyclic graph Shubo Chen 1, Fangli Xia 1, Xia Cai, Jianguang

More information

The expected values of Kirchhoff indices in the random polyphenyl and spiro chains

The expected values of Kirchhoff indices in the random polyphenyl and spiro chains Also available at http://amc-journal.eu ISSN 1855-39 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 9 (015) 197 07 The expected values of Kirchhoff indices in the random

More information

Paths and walks in acyclic structures: plerographs versus kenographs

Paths and walks in acyclic structures: plerographs versus kenographs Issue in onor of Prof. Alexander Balaban ARKIVO 2005 (x) 33-44 Paths and walks in acyclic structures: plerographs versus kenographs Damir Vukičević,* a Ante Miličević, b Sonja Nikolić, b Jelena Sedlar,

More information

The smallest Randić index for trees

The smallest Randić index for trees Proc. Indian Acad. Sci. (Math. Sci.) Vol. 123, No. 2, May 2013, pp. 167 175. c Indian Academy of Sciences The smallest Randić index for trees LI BINGJUN 1,2 and LIU WEIJUN 2 1 Department of Mathematics,

More information

On the Inverse Sum Indeg Index

On the Inverse Sum Indeg Index On the Inverse Sum Indeg Index Jelena Sedlar a, Dragan Stevanović b,c,, Alexander Vasilyev c a University of Split, Faculty of Civil Engineering, Matice Hrvatske 15, HR-21000 Split, Croatia b Mathematical

More information

The Eccentric Connectivity Index of Dendrimers

The Eccentric Connectivity Index of Dendrimers Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 45, 2231-2236 The Eccentric Connectivity Index of Dendrimers Jianguang Yang and Fangli Xia Department of Mathematics, Hunan City University Yiyang, Hunan

More information

Applications of Isometric Embeddings to Chemical Graphs

Applications of Isometric Embeddings to Chemical Graphs DIMACS Series in Discrete Mathematics and Theoretical Computer Science Applications of Isometric Embeddings to Chemical Graphs Sandi Klavžar Abstract. Applications of isometric embeddings of benzenoid

More information

Saturation number of lattice animals

Saturation number of lattice animals ISSN 1855-966 (printed edn.), ISSN 1855-974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (018) 191 04 https://doi.org/10.649/1855-974.179.bce (Also available at http://amc-journal.eu) Saturation

More information

Generalized Wiener Indices in Hexagonal Chains

Generalized Wiener Indices in Hexagonal Chains Generalized Wiener Indices in Hexagonal Chains Sen-Peng Eu Dept. of Mathematics Nat l U. of Kaohsiung Kaohsiung, Taiwan speu@nuk.edu.tw Bo-Yin Yang Dept. of Mathematics Tamkang University Tamsui, Taiwan

More information

Computing Banhatti Indices of Hexagonal, Honeycomb and Derived Networks

Computing Banhatti Indices of Hexagonal, Honeycomb and Derived Networks American Journal of Mathematical and Computer Modelling 018; 3(): 38-45 http://www.sciencepublishinggroup.com/j/ajmcm doi: 10.11648/j.ajmcm.018030.11 Computing Banhatti Indices of Hexagonal, Honeycomb

More information

CCO Commun. Comb. Optim.

CCO Commun. Comb. Optim. Communications in Combinatorics and Optimization Vol. 3 No., 018 pp.179-194 DOI: 10.049/CCO.018.685.109 CCO Commun. Comb. Optim. Leap Zagreb Indices of Trees and Unicyclic Graphs Zehui Shao 1, Ivan Gutman,

More information

arxiv: v1 [math.co] 15 Sep 2016

arxiv: v1 [math.co] 15 Sep 2016 A Method for Computing the Edge-Hyper-Wiener Index of Partial Cubes and an Algorithm for Benzenoid Systems Niko Tratnik Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia arxiv:1609.0469v1

More information

THE HARMONIC INDEX OF SUBDIVISION GRAPHS. Communicated by Ali Reza Ashrafi. 1. Introduction

THE HARMONIC INDEX OF SUBDIVISION GRAPHS. Communicated by Ali Reza Ashrafi. 1. Introduction Transactions on Combinatorics ISSN print: 5-8657, ISSN on-line: 5-8665 Vol. 6 No. 07, pp. 5-7. c 07 University of Isfahan toc.ui.ac.ir www.ui.ac.ir THE HARMONIC INDEX OF SUBDIVISION GRAPHS BIBI NAIME ONAGH

More information

Research Article. Generalized Zagreb index of V-phenylenic nanotubes and nanotori

Research Article. Generalized Zagreb index of V-phenylenic nanotubes and nanotori Available online www.jocpr.com Journal of Chemical and Pharmaceutical Research, 2015, 7(11):241-245 Research Article ISSN : 0975-7384 CODEN(USA) : JCPRC5 Generalized Zagreb index of V-phenylenic nanotubes

More information

Improved bounds on the difference between the Szeged index and the Wiener index of graphs

Improved bounds on the difference between the Szeged index and the Wiener index of graphs Improved bounds on the difference between the Szeged index and the Wiener index of graphs Sandi Klavžar a,b,c M. J. Nadjafi-Arani d October 5, 013 a Faculty of Mathematics and Physics, University of Ljubljana,

More information

BoundsonVertexZagrebIndicesofGraphs

BoundsonVertexZagrebIndicesofGraphs Global Journal of Science Frontier Research: F Mathematics and Decision Sciences Volume 7 Issue 6 Version.0 Year 07 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals

More information

Extremal trees with fixed degree sequence for atom-bond connectivity index

Extremal trees with fixed degree sequence for atom-bond connectivity index Filomat 26:4 2012), 683 688 DOI 10.2298/FIL1204683X Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Extremal trees with fixed degree

More information

LAPLACIAN ESTRADA INDEX OF TREES

LAPLACIAN ESTRADA INDEX OF TREES MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 63 (009) 769-776 ISSN 0340-653 LAPLACIAN ESTRADA INDEX OF TREES Aleksandar Ilić a and Bo Zhou b a Faculty

More information

Some properties of the first general Zagreb index

Some properties of the first general Zagreb index AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 47 (2010), Pages 285 294 Some properties of the first general Zagreb index Muhuo Liu Bolian Liu School of Mathematic Science South China Normal University Guangzhou

More information

Some Computational Aspects for the Line Graph of Banana Tree Graph

Some Computational Aspects for the Line Graph of Banana Tree Graph Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 6 (017), pp. 601-67 Research India Publications http://www.ripublication.com/gjpam.htm Some Computational Aspects for the

More information

ON THE WIENER INDEX AND LAPLACIAN COEFFICIENTS OF GRAPHS WITH GIVEN DIAMETER OR RADIUS

ON THE WIENER INDEX AND LAPLACIAN COEFFICIENTS OF GRAPHS WITH GIVEN DIAMETER OR RADIUS MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 63 (2010) 91-100 ISSN 0340-6253 ON THE WIENER INDEX AND LAPLACIAN COEFFICIENTS OF GRAPHS WITH GIVEN DIAMETER

More information

THE GENERALIZED ZAGREB INDEX OF CAPRA-DESIGNED PLANAR BENZENOID SERIES Ca k (C 6 )

THE GENERALIZED ZAGREB INDEX OF CAPRA-DESIGNED PLANAR BENZENOID SERIES Ca k (C 6 ) Open J Math Sci, Vol 1(2017), No 1, pp 44-51 ISSN 2523-0212 Website: http://wwwopenmathsciencecom THE GENERALIZED ZAGREB INDEX OF CAPRA-DESIGNED PLANAR BENZENOID SERIES Ca k (C 6 ) MUHAMMAD S SARDAR, SOHAIL

More information

K-DOMINATION ON HEXAGONAL CACTUS CHAINS

K-DOMINATION ON HEXAGONAL CACTUS CHAINS Kragujevac Journal of Mathematics Volume 36 Number (01), Pages 335 347. K-DOMINATION ON HEXAGONAL CACTUS CHAINS SNJE ZANA MAJSTOROVIĆ 1, TOMISLAV DO SLIĆ, AND ANTOANETA KLOBU CAR 3 Abstract. In this paper

More information

On the Randić index. Huiqing Liu. School of Mathematics and Computer Science, Hubei University, Wuhan , China

On the Randić index. Huiqing Liu. School of Mathematics and Computer Science, Hubei University, Wuhan , China Journal of Mathematical Chemistry Vol. 38, No. 3, October 005 ( 005) DOI: 0.007/s090-005-584-7 On the Randić index Huiqing Liu School of Mathematics and Computer Science, Hubei University, Wuhan 43006,

More information

EXACT FORMULAE OF GENERAL SUM-CONNECTIVITY INDEX FOR SOME GRAPH OPERATIONS. Shehnaz Akhter, Rashid Farooq and Shariefuddin Pirzada

EXACT FORMULAE OF GENERAL SUM-CONNECTIVITY INDEX FOR SOME GRAPH OPERATIONS. Shehnaz Akhter, Rashid Farooq and Shariefuddin Pirzada MATEMATIČKI VESNIK MATEMATIQKI VESNIK 70, 3 2018, 267 282 September 2018 research paper originalni nauqni rad EXACT FORMULAE OF GENERAL SUM-CONNECTIVITY INDEX FOR SOME GRAP OPERATIONS Shehnaz Akhter, Rashid

More information

Graphs with maximum Wiener complexity

Graphs with maximum Wiener complexity Graphs with maximum Wiener complexity Yaser Alizadeh a Sandi Klavžar b,c,d a Department of Mathematics, Hakim Sabzevari University, Sabzevar, Iran e-mail: y.alizadeh@hsu.ac.ir b Faculty of Mathematics

More information

arxiv: v1 [math.co] 12 Jul 2011

arxiv: v1 [math.co] 12 Jul 2011 On augmented eccentric connectivity index of graphs and trees arxiv:1107.7v1 [math.co] 1 Jul 011 Jelena Sedlar University of Split, Faculty of civil engeneering, architecture and geodesy, Matice hrvatske

More information

Kragujevac J. Sci. 33 (2011) UDC : ZAGREB POLYNOMIALS OF THORN GRAPHS. Shuxian Li

Kragujevac J. Sci. 33 (2011) UDC : ZAGREB POLYNOMIALS OF THORN GRAPHS. Shuxian Li Kragujevac J. Sci. 33 (2011) 33 38. UDC 541.27:541.61 ZAGREB POLYNOMIALS OF THORN GRAPHS Shuxian Li Department of Mathematics, South China Normal University, Guangzhou 510631, China e-mail: lishx1002@126.com

More information

THE WIENER INDEX AND HOSOYA POLYNOMIAL OF A CLASS OF JAHANGIR GRAPHS

THE WIENER INDEX AND HOSOYA POLYNOMIAL OF A CLASS OF JAHANGIR GRAPHS Fundamental Journal of Mathematics and Mathematical Sciences Vol. 3, Issue, 05, Pages 9-96 This paper is available online at http://www.frdint.com/ Published online July 3, 05 THE WIENER INDEX AND HOSOYA

More information

It can be easily shown that the above definition is equivalent to

It can be easily shown that the above definition is equivalent to DOI 10.1186/s40064-016-1864-7 RESEARCH Open Access The F coindex of some graph operations Nilanjan De 1*, Sk. Md. Abu Nayeem 2 and Anita Pal 3 *Correspondence: de.nilanjan@rediffmail.com 1 Department of

More information

On the harmonic index of the unicyclic and bicyclic graphs

On the harmonic index of the unicyclic and bicyclic graphs On the harmonic index of the unicyclic and bicyclic graphs Yumei Hu Tianjin University Department of Mathematics Tianjin 30007 P. R. China huyumei@tju.edu.cn Xingyi Zhou Tianjin University Department of

More information

A NOVEL/OLD MODIFICATION OF THE FIRST ZAGREB INDEX

A NOVEL/OLD MODIFICATION OF THE FIRST ZAGREB INDEX A NOVEL/OLD MODIFICATION OF THE FIRST ZAGREB INDEX AKBAR ALI 1 AND NENAD TRINAJSTIĆ arxiv:1705.1040v1 [math.co] 0 May 017 Abstract. In the paper [Gutman, I.; Trinajstić, N. Chem. Phys. Lett. 197, 17, 55],

More information

Relation Between Randić Index and Average Distance of Trees 1

Relation Between Randić Index and Average Distance of Trees 1 MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 66 (011) 605-61 ISSN 0340-653 Relation Between Randić Index and Average Distance of Trees 1 Marek Cygan,

More information

A Method of Calculating the Edge Szeged Index of Hexagonal Chain

A Method of Calculating the Edge Szeged Index of Hexagonal Chain MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 68 (2012) 91-96 ISSN 0340-6253 A Method of Calculating the Edge Szeged Index of Hexagonal Chain Shouzhong

More information

M-polynomials and Degree based Topological Indices for the Line Graph of Firecracker Graph

M-polynomials and Degree based Topological Indices for the Line Graph of Firecracker Graph Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 6 (017), pp. 749-776 Research India Publications http://www.ripublication.com/gjpam.htm M-polynomials and Degree based Topological

More information

Estimating Some General Molecular Descriptors of Saturated Hydrocarbons

Estimating Some General Molecular Descriptors of Saturated Hydrocarbons Estimating Some General Molecular Descriptors of Saturated Hydrocarbons Akbar Ali, Zhibin Du, Kiran Shehzadi arxiv:1812.11115v1 [math.co] 28 Dec 2018 Knowledge Unit of Science, University of Management

More information

Calculating the hyper Wiener index of benzenoid hydrocarbons

Calculating the hyper Wiener index of benzenoid hydrocarbons Calculating the hyper Wiener index of benzenoid hydrocarbons Petra Žigert1, Sandi Klavžar 1 and Ivan Gutman 2 1 Department of Mathematics, PEF, University of Maribor, Koroška 160, 2000 Maribor, Slovenia

More information

The Story of Zagreb Indices

The Story of Zagreb Indices The Story of Zagreb Indices Sonja Nikolić CSD 5 - Computers and Scientific Discovery 5 University of Sheffield, UK, July 0--3, 00 Sonja Nikolic sonja@irb.hr Rugjer Boskovic Institute Bijenicka cesta 54,

More information

Interpolation method and topological indices: 2-parametric families of graphs

Interpolation method and topological indices: 2-parametric families of graphs Interpolation method and topological indices: 2-parametric families of graphs Yaser Alizadeh Department of Mathematics, Tarbiat Modares University, Tehran, Iran e-mail: yaser alizadeh s@yahoo.com Sandi

More information

Harary Index and Hamiltonian Property of Graphs

Harary Index and Hamiltonian Property of Graphs MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 70 (2013) 645-649 ISSN 0340-6253 Harary Index and Hamiltonian Property of Graphs Ting Zeng College of Mathematics

More information

Wiener index of generalized 4-stars and of their quadratic line graphs

Wiener index of generalized 4-stars and of their quadratic line graphs AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 58() (204), Pages 9 26 Wiener index of generalized 4-stars and of their quadratic line graphs Martin Knor Slovak University of Technology in Bratislava Faculty

More information

arxiv: v1 [math.co] 14 May 2018

arxiv: v1 [math.co] 14 May 2018 On Neighbourhood Zagreb index of product graphs Sourav Mondal a, Nilanjan De b, Anita Pal c arxiv:1805.05273v1 [math.co] 14 May 2018 Abstract a Department of mathematics, NIT Durgapur, India. b Department

More information

The Linear Chain as an Extremal Value of VDB Topological Indices of Polyomino Chains

The Linear Chain as an Extremal Value of VDB Topological Indices of Polyomino Chains Applied Mathematical Sciences, Vol. 8, 2014, no. 103, 5133-5143 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.46507 The Linear Chain as an Extremal Value of VDB Topological Indices of

More information

The multiplicative Zagreb indices of graph operations

The multiplicative Zagreb indices of graph operations Das et al Journal of Inequalities and Applications 0, 0:90 R E S E A R C H Open Access The multiplicative Zagreb indices of graph operations Kinkar C Das, Aysun Yurttas,MugeTogan, Ahmet Sinan Cevik and

More information

arxiv: v1 [math.co] 3 Jul 2017

arxiv: v1 [math.co] 3 Jul 2017 ON THE EXTREMAL GRAPHS WITH RESPECT TO BOND INCIDENT DEGREE INDICES AKBAR ALI a AND DARKO DIMITROV b arxiv:1707.00733v1 [math.co] 3 Jul 2017 Abstract. Many existing degree based topological indices can

More information

MULTIPLICATIVE ZAGREB ECCENTRICITY INDICES OF SOME COMPOSITE GRAPHS. Communicated by Ali Reza Ashrafi. 1. Introduction

MULTIPLICATIVE ZAGREB ECCENTRICITY INDICES OF SOME COMPOSITE GRAPHS. Communicated by Ali Reza Ashrafi. 1. Introduction Transactions on Combinatorics ISSN (print): 2251-8657, ISSN (on-line): 2251-8665 Vol. 3 No. 2 (2014), pp. 21-29. c 2014 University of Isfahan www.combinatorics.ir www.ui.ac.ir MULTIPLICATIVE ZAGREB ECCENTRICITY

More information

CALCULATING THE DEGREE DISTANCE OF PARTIAL HAMMING GRAPHS

CALCULATING THE DEGREE DISTANCE OF PARTIAL HAMMING GRAPHS CALCULATING THE DEGREE DISTANCE OF PARTIAL HAMMING GRAPHS Aleksandar Ilić Faculty of Sciences and Mathematics, University of Niš, Serbia e-mail: aleksandari@gmail.com Sandi Klavžar Faculty of Mathematics

More information

On trees with smallest resolvent energies 1

On trees with smallest resolvent energies 1 On trees with smallest resolvent energies 1 Mohammad Ghebleh, Ali Kanso, Dragan Stevanović Faculty of Science, Kuwait University, Safat 13060, Kuwait mamad@sci.kuniv.edu.kw, akanso@sci.kuniv.edu.kw, dragance106@yahoo.com

More information

About Atom Bond Connectivity and Geometric-Arithmetic Indices of Special Chemical Molecular and Nanotubes

About Atom Bond Connectivity and Geometric-Arithmetic Indices of Special Chemical Molecular and Nanotubes Journal of Informatics and Mathematical Sciences Vol. 10, Nos. 1 & 2, pp. 153 160, 2018 ISSN 0975-5748 (online); 0974-875X (print) Published by RGN Publications http://dx.doi.org/10.26713/jims.v10i1-2.545

More information

On the Randić Index of Polyomino Chains

On the Randić Index of Polyomino Chains Applied Mathematical Sciences, Vol. 5, 2011, no. 5, 255-260 On the Randić Index of Polyomino Chains Jianguang Yang, Fangli Xia and Shubo Chen Department of Mathematics, Hunan City University Yiyang, Hunan

More information

THE HARMONIC INDEX OF UNICYCLIC GRAPHS WITH GIVEN MATCHING NUMBER

THE HARMONIC INDEX OF UNICYCLIC GRAPHS WITH GIVEN MATCHING NUMBER Kragujevac Journal of Mathematics Volume 38(1 (2014, Pages 173 183. THE HARMONIC INDEX OF UNICYCLIC GRAPHS WITH GIVEN MATCHING NUMBER JIAN-BO LV 1, JIANXI LI 1, AND WAI CHEE SHIU 2 Abstract. The harmonic

More information

THE REVISED EDGE SZEGED INDEX OF BRIDGE GRAPHS

THE REVISED EDGE SZEGED INDEX OF BRIDGE GRAPHS Hacettepe Journal of Mathematics and Statistics Volume 1 01 559 566 THE REVISED EDGE SZEGED INDEX OF BRIDGE GRAPHS Hui Dong and Bo Zhou Received 01:09:011 : Accepted 15:11:011 e=uv EG Abstract The revised

More information

A lower bound for the harmonic index of a graph with minimum degree at least two

A lower bound for the harmonic index of a graph with minimum degree at least two Filomat 7:1 013), 51 55 DOI 10.98/FIL1301051W Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat A lower bound for the harmonic index

More information

Clar number of catacondensed benzenoid hydrocarbons

Clar number of catacondensed benzenoid hydrocarbons Clar number of catacondensed benzenoid hydrocarbons Sandi Klavžar a,, Petra Žigert a, Ivan Gutman b, a Department of Mathematics, PEF, University of Maribor, Koroška 160, 2000 Maribor, Slovenia b Faculty

More information

M-POLYNOMIALS AND DEGREE-BASED TOPOLOGICAL INDICES OF SOME FAMILIES OF CONVEX POLYTOPES

M-POLYNOMIALS AND DEGREE-BASED TOPOLOGICAL INDICES OF SOME FAMILIES OF CONVEX POLYTOPES Open J. Math. Sci., Vol. 2(2018), No. 1, pp. 18-28 ISSN 2523-0212 Website: http://www.openmathscience.com M-POLYNOMIALS AND DEGREE-BASED TOPOLOGICAL INDICES OF SOME FAMILIES OF CONVEX POLYTOPES MUHAMMAD

More information

The Wiener Polynomial of a Graph

The Wiener Polynomial of a Graph The Wiener Polynomial of a Graph Bruce E. Sagan 1 Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, U.S.A. Yeong-Nan Yeh 2 Institute of Mathematics, Academia Sinica, Nankang,

More information

The eccentric-distance sum of some graphs

The eccentric-distance sum of some graphs Electronic Journal of Graph Theory an Applications 5 (1) (017), 51 6 The eccentric-istance sum of some graphs Pamapriya P., Veena Matha Department of Stuies in Mathematics University of Mysore, Manasagangotri

More information

The second maximal and minimal Kirchhoff indices of unicyclic graphs 1

The second maximal and minimal Kirchhoff indices of unicyclic graphs 1 MATCH Communications in Mathematica and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 61 (009) 683-695 ISSN 0340-653 The second maxima and minima Kirchhoff indices of unicycic graphs 1 Wei Zhang,

More information

arxiv: v1 [math.co] 17 Feb 2017

arxiv: v1 [math.co] 17 Feb 2017 Properties of the Hyper-Wiener index as a local function arxiv:170.05303v1 [math.co] 17 Feb 017 Ya-Hong Chen 1,, Hua Wang 3, Xiao-Dong Zhang 1 1 Department of Mathematics, Lishui University Lishui, Zhejiang

More information

A class of trees and its Wiener index.

A class of trees and its Wiener index. A class of trees and its Wiener index. Stephan G. Wagner Department of Mathematics Graz University of Technology Steyrergasse 3, A-81 Graz, Austria wagner@finanz.math.tu-graz.ac.at Abstract In this paper,

More information

arxiv: v1 [math.co] 5 Sep 2016

arxiv: v1 [math.co] 5 Sep 2016 Ordering Unicyclic Graphs with Respect to F-index Ruhul Amin a, Sk. Md. Abu Nayeem a, a Department of Mathematics, Aliah University, New Town, Kolkata 700 156, India. arxiv:1609.01128v1 [math.co] 5 Sep

More information

A Note on Revised Szeged Index of Graph Operations

A Note on Revised Szeged Index of Graph Operations Iranian J Math Che 9 (1) March (2018) 57 63 Iranian Journal of Matheatical Cheistry Journal hoepage: ijckashanuacir A Note on Revised Szeged Index of Graph Operations NASRIN DEHGARDI Departent of Matheatics

More information

ON SOME DEGREE BASED TOPOLOGICAL INDICES OF T io 2 NANOTUBES

ON SOME DEGREE BASED TOPOLOGICAL INDICES OF T io 2 NANOTUBES U.P.B. Sci. Bull., Series B, Vol. 78, Iss. 4, 2016 ISSN 1454-2331 ON SOME DEGREE BASED TOPOLOGICAL INDICES OF T io 2 NANOTUBES Abdul Qudair Baig 1 and Mehar Ali Malik 2 Two well known connectivity topological

More information

Mathematical model for determining octanol-water partition coeffi cient of alkanes

Mathematical model for determining octanol-water partition coeffi cient of alkanes Mathematical model for determining octanol-water partition coeffi cient of alkanes Jelena Sedlar University of Split, Croatia 2nd Croatian Combinatorial Days, Zagreb, Croatia September 27-28, 2018 Jelena

More information

A note on the Laplacian Estrada index of trees 1

A note on the Laplacian Estrada index of trees 1 MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 63 (2009) 777-782 ISSN 0340-6253 A note on the Laplacian Estrada index of trees 1 Hanyuan Deng College of

More information

ON THE DISTANCE-BASED TOPOLOGICAL INDICES OF FULLERENE AND FULLERENYL ANIONS HAVING DENDRIMER UNITS

ON THE DISTANCE-BASED TOPOLOGICAL INDICES OF FULLERENE AND FULLERENYL ANIONS HAVING DENDRIMER UNITS Digest Journal of Nanomaterials and Biostructures Vol. 4, No. 4, December 2009, p. 713 722 ON THE DISTANCE-BASED TOPOLOGICAL INDICES OF FULLERENE AND FULLERENYL ANIONS HAVING DENDRIMER UNITS MAJID AREZOOMAND

More information