Short communication A New Version of Atom-Bond Connectivity Index Ante Graovac1 and Modjtaba Ghorbani2* 1 Institute »R. Boskovic«, HR-10002 Zagreb, POB 180, Croatia, and Faculty of Science, University of Split, Nikole Tesle 12, HR-21000, Split, Croatia 2 Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785-136, I. R. Iran * Corresponding author: E-mail: mghorbani@srttu.edu Received: 12-03-2010 This paper is dedicated to Professor Milan Randi} on the occasion of his 80th birthday Abstract , HEI The atom-bond connectivity index is a recently introduced topological index defined as , where du ABC,(G)- z EHH denotes degree of vertex u. Here we define a new version of the ABC index as '""' " " , where nu denotes the number of vertices of G whose distances to vertex u are smaller than those to other vertex v of the edge e = uv, and nv is defined analogously. The goal of this paper is to study the ABC2 index. Keywords: Topological indices, ABC Index, ABC2 Index. 1. Introduction Mathematical chemistry is a branch of theoretical chemistry using mathematical methods to discuss and predict molecular properties without necessarily referring to quantum mechanics.1-3 Chemical graph theory is a branch of mathematical chemistry which applies graph theory in mathematical modeling of chemical phenomena.4 This theory has an important effect on the development of the chemical sciences. A graph is a collection of points and lines connecting them. The points and lines of a graph are also called vertices and edges respectively. If e is an edge of G, connecting the vertices u and v, then we write e = uv and say »u and v are adjacent«. A connected graph is a graph such that there exists a path between all pairs of vertices. The distance d(u,v) = dG(u,v) between two vertices u and v is the length of the shortest path between u and v in G. A simple graph is an unweighted, undirected graph without loops or multiple edges. A molecular graph is a simple graph such that its vertices correspond to the atoms and the edges to the bonds. Note that hydrogen atoms are often omitted. According to the IUPAC terminology, a topological index is a numerical value associated with chemical constitution which can be then used for correlation of chemical structure with various physical and chemical properties, chemical reactivity and biological activity.5-12 Let X be the class of finite graphs. A topological index is a function Top from X into real numbers where for G and H being isomorphic: Top(G) = Top(H). Obviously, the number of vertices and the number of edges are topo-logical indices. The Wiener index is the first graph invariant reported (distance based) topological index and is defined as a half sum of the distances between all the pairs of vertices in a molecular graph. Let G be a connected graph and e = uv be an edge of G. The number of vertices of G whose distance to the vertex u is smaller than the distance to the vertex v is denoted by nu = nu(e\G). Analogously, nv = nv(e\G) is the number of vertices of G whose distance to the vertex v is smaller than to u. The vertex Szeged index is another topological index which was introduced by Gutman.10 It is defined by: Sz (G) = X nu (e)nv(e). The edge Szeged index of G is a recently proposed topologic al index11 defined as Sz (G) = X mu (e)mv(e). where mu = mu(e\G) (and mv= mv(e|G)= "denSie the number of edges of G whose distances to the vertex u are smaller than those to v (the number of edges of G whose distances to the vertex v are smaller than those to u). Motivated by the success of the vertex Szeged index, Khadikar et al.12' 13 proposed a seemingly similar molecular structure descriptor that in what follows we call the edge-PI index. In analogy with definition of the vertex Szeged index, the edge-PI index is defined as Pl(G) = X[mu (e) + mv(e)]. Quite recently the vertex-version of the PI index was also considered.14 It is defined as Plv(G) = X [mu (e | G) + nv(e | G)]The atom-bond connectivity indeex is a novel topological index and was defined by Estrada et. al.15 as where du stands for the degree of vertex u. Now we define a new version of the atom-bond connectivity index as ABC\{G)~ X e=iii'e£(G > v n» The goal of this paper is to study the ABC2 index. Our notation is standard and mainly taken from standard books of chemical graph theory.5 All graphs considered in this paper are finite, undirected, simple and connected. For background materials, see references.16-22 2. Results and Discussions In this section we first determine some bounds for ABC2(G) index. Next we introduce the notion of transitive and edge-transitive action on vertices of graph G. Finally, by using this concept and some Lemmas we compute the ABC2 index of the hypercube graph. An automorphism of the graph G = (V E) is a bijec-tion o on V which preserves the edge set E, i. e., if e = uv is an edge, then o(e) = o(u)o(v) is an edge of E. Here the image of vertex u under o is denoted by o(u). The set of all automorphisms of G under the composition of mappings forms a group which is denoted by Aut(G). Aut(G) acts transitively on V if for any vertices u and v in V there is a e Aut(G) = such that a(u) = v. Similarly G = (V, E) is called an edge-transitive graph if for any two edges e1 = uv and e2 = xy in E there is an element ß e Aut(G) such that ß(ej = e2 where ß(e:) = ß(u)ß(v). Lemma 3. If G is edge transitive, then ABC(G) =|£| EBZ1 v « for any e = uv e E (G). Lemma 4. If G is edge transitive, then ~-2 ABC2(G)=\E I 1"» +",- ~ V «LA , for any e = uv e E (G). Example 5. Let Sn be the star graph with n + 1 vertices. It is easy to see that Sn is edge -transitive. Also, Sn is a tree and so by using Lemma 5 we have: ABC2(S„) = n = Mn-l), Fullerenes are molecules in the form of polyhedral closed cages made up entirely of n three-coordinated carbon atoms and having 12 pentagonal and (n/2 - 10) hexagonal faces, where n is an even number equal or greater than 20. Hence, the smallest fullerene, C20 (n = 20), has 12 pentagons. In the following example we compute the ABC2 index of C20. Example 6. Consider the fullerene graph C20 shown in figure 1. It is easy to see C20 is edge-transitive and so by computing values of nu and nv, we have nu = nv = 8. Therefore: |E| = 30 and Figure 1: The graph of fullerene C20. The fullerene C20 is the only edge-transitive fullerene. So it is important to be able to compute ABC2 index in the case where G is not an edge-transitive graph. One can apply then the following Lemma: Lemma 1. Let G = (V E) be a graph. If Aut(G) on E has orbits E, 1 < i < s, where ei= utvt is an edge of E, then: \d„+dr- 2 ABC(G) = XI Ei\J ' ' m V d.. -d„ and Proof. The values of nu's for every e = uv e Ei are equal. So, it is enough to compute nu and nv for ei = uv (1 < i < s). Example 2. Let Pn be the path on n vertices. Pn is not edge - transitive and by using Lemma 8 we have: ABC1(Pn)=^Z2[ 1 1 •Jn-l yj2(>t-2) 1 j. V3(«-3) -Jn^l A hypercube is defined as follows. The vertex set of the hypercube Hnconsists of all n-tuples b1b2...bn with bt e{0,1). Two vertices are adjacent if the corresponding tuples differ in precisely one place. Darafsheh23 proved Hn is vertex and edge transitive. He also computed nu and nv for every edge e = uv as nu = nv = 2n-1. By using this result we have the following: Theorem 7. Let Hn be the hypercube graph. Then, Theorem 8. Let G = (V E) be a graph. Then ABC2(G) < Plv(G) - 2 | E | and Proof. By the definition of ABC2(G) index it is easy to see that ABC2(G) < I Jnu + nv- 2< I nu + nv - 2 = 2 e = uv u v e = uv u v Plv(G) - 2 | E |. This determines the upper bound. For the lower bound one can see that: n +„ _2 2 n„+nv-2 [ABC,{G)f i I ' ' ~ I n„Jiv Theorem 9. Let G be a bipartite graph on n vertices. Then we have: ^L^s]n~2 1. So, abc2(G)< i V i Also, for every edge e = uv, nu nv < n2 / 4. Corollary 10. Let T be a tree with n vertices. Then we have: 2t"~')V/T-2