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On Harary Index of Graphs✩ CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Discrete Applied Mathematics 159 (2011) 1631–1640 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam On Harary index of graphsI Kexiang Xu a, Kinkar Ch. Das b,∗ a College of Science, Nanjing University of Aeronautics & Astronautics, Nanjing, PR China b Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea article info a b s t r a c t Article history: The Harary index is defined as the sum of reciprocals of distances between all pairs of Received 9 November 2010 vertices of a connected graph. For a connected graph G D .V ; E/ and two nonadjacent Received in revised form 20 April 2011 vertices vi and vj in V .G/ of G, recall that GCvivj is the supergraph formed from G by adding Accepted 5 June 2011 an edge between vertices v and v . Denote the Harary index of G and G C v v by H.G/ and Available online 8 July 2011 i j i j H.G C vivj/, respectively. We obtain lower and upper bounds on H.G C vivj/ − H.G/, and characterize the equality cases in those bounds. Finally, in this paper, we present some Keywords: lower and upper bounds on the Harary index of graphs with different parameters, such as Graph Diameter clique number and chromatic number, and characterize the extremal graphs at which the Harary index lower or upper bounds on the Harary index are attained. Clique number ' 2011 Elsevier B.V. All rights reserved. Chromatic number 1. Introduction The Harary index of a graph G, denoted by H.G/, has been introduced independently by Plav²i¢ et al. [24] and by Ivanciuc et al. [19] in 1993. It has been named in honor of Professor Frank Harary on the occasion of his 70th birthday. The Harary index is defined as follows: X 1 H D H.G/ D d .u; v/ u;v2V .G/ G where the summation goes over all unordered pairs of vertices of G and dG.u; v/ denotes the distance of the two vertices u and v in the graph G (i.e., the number of edges in a shortest path connecting u and v). Mathematical properties and applications of H are reported in [4,8,9,23,29–31,33]. Another two related distance-based topological indices of the graph G are the Wiener index W .G/ and the hyper-Wiener index WW .G/. As an oldest topological index, the Wiener index of a (molecular) graph G, first introduced by Wiener [28] in 1947, was defined as X W .G/ D dG.u; v/ u;v2V .G/ I The first author is supported by NUAA Research Founding, No. NS2010205, the second author is supported by BK21 Math Modeling HRD Div. Sungkyunkwan University, Suwon, Republic of Korea. ∗ Corresponding author. Fax: +82 31 290 7033. E-mail addresses: [email protected] (K. Xu), [email protected], [email protected] (K.Ch. Das). 0166-218X/$ – see front matter ' 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.dam.2011.06.003 1632 K. Xu, K.Ch. Das / Discrete Applied Mathematics 159 (2011) 1631–1640 with the summation going over all unordered pairs of vertices of G. The hyper-Wiener index of G, first introduced by Randi¢ [25] in 1993, is nowadays defined as [22]: 1 X 1 X 2 WW .G/ D dG.u; v/ C dG.u; v/ : 2 2 u;v2V .G/ u;v2V .G/ Mathematical properties and applications of Wiener index and hyper-Wiener index are extensively reported in the literature [6,7,9–16,28]. Let γ .G; k/ be the number of vertex pairs of the graph G that are at distance k. Then X 1 H.G/ D γ .G; k/: (1) k k≥1 The maximum value of k for which γ .G; k/ is non-zero, is the diameter of the graph G, and will be denoted by d. Note that, in any disconnected graph G, the distance is infinite of any two vertices from two distinct components. Therefore its reciprocal can be viewed as 0. Thus, we can define validly the Harary index of disconnected graph G as follows: Xk H.G/ D H.Gi/ where G1; G2;:::; Gk are the components of G: iD1 A molecular graph is a connected graph of maximum degree at most 4. It models the skeleton of a molecule [27]. The bounds of a descriptor are important information of a (molecular) graph in the sense that they establish the approximate range of the descriptor in terms of molecular structural parameters. Since determining the exact value of the Harary index of a (molecular) graph, though this formula (1) is efficient, is generally difficult, providing some (upper or lower) bounds for the Harary index of some (molecular) graphs is very useful. Its applications in this aspect includes the characterization of molecular graphs, measuring landscape connectivity [20,26], etc. (see [5,21,33] and the references therein). All graphs considered in this paper are finite and simple. Let G be a graph with vertex set V .G/ D fv1; v2; : : : ; vng and edge set E.G/. The degree of vi 2 V .G/, denoted by dG.vi/ or d.vi/ for short, is the number of vertices in G adjacent to vi. For a subset W of V .G/, let G − W be the subgraph of G obtained by deleting the vertices of W and the edges incident with them. 0 0 0 Similarly, for a subset E of E.G/, we denote by G − E the subgraph of G obtained by deleting the edges of E . If W D fvig 0 0 and E D fvjvkg, the subgraphs G − W and G − E will be written as G − vi and G − vjvk for short, respectively. For any two nonadjacent vertices vi and vj in graph G, we use G C vivj to denote the graph obtained by adding a new edge vivj to graph G. The chromatic number of a graph G, denoted by χ.G/, is the minimum number of colors such that G can be colored with these colors such that no two adjacent vertices have the same color. A clique of graph G is a subset V0 of V .G/ such that in GTV0U, the subgraph of G induced by V0, any two vertices are adjacent. The clique number of G, denoted by !.G/, is the number of vertices in the largest clique of G. We denote by Kn; Pn and Cn the complete graph, the path and the cycle on n vertices throughout this paper. For other undefined notations and terminology from graph theory, the readers are referred to [1]. Let Wn;k and Xn;k be the set of connected n-vertex graphs with clique number k and the set of connected graphs of order C C···C D n with chromatic number k, respectively. Assume that n1 n2 nk n. We denote by Kn1;n2;:::;nk the complete k-partite graph of order n whose partition sets are of size n1; n2;:::; nk, respectively. Recall that, as a special k-partite graph of order n, the Turán graph Tn.k/ is a complete k-partite graph whose partition sets differ in size by at most 1. Denote by Kn;k the graph obtained by identifying one vertex of Kk with a pendent vertex of path Pn−kC1. Note that Kn;k is just a kite graph. The paper is organized as follows. In Section 2, we give a list of lemmas and preliminaries. In Section 3, we obtain lower and upper bounds on H.G C vivj/ − H.G/, and characterize the equality cases in those bounds, where vi and vj are two nonadjacent vertices of G. Finally, in Section 4, we present some lower and upper bounds on the Harary index of graphs with different parameters, such as clique number and chromatic number, and characterize the extremal graphs at which the lower or upper bounds on the Harary index are attained, in particular, the extremal graphs in Wn;k and Xn;k are completely characterized, respectively. 2. Preliminaries In this section, we list or prove some lemmas as basic but necessary preliminaries, which will be used in the subsequent proofs. P dG.vi;vj/ x For a graph G with vi 2 V .G/, we define QG.vi/ D . Note that the function f .x/ D is strictly vj2V .G/ dG.vi;vj/C1 xC1 increasing for x ≥ 1. Since the following two lemmas are from an unpublished paper [29] (which is under review in some journal), we give the proofs below for them. Lemma 2.1 ([29]). Let G be a graph of order n and vi be a pendent vertex of G with vivj 2 E.G/. Then we have H.G/ D − C − − H.G vi/ n 1 QG−vi .vj/. K. Xu, K.Ch. Das / Discrete Applied Mathematics 159 (2011) 1631–1640 1633 Fig. 1. The graphs Gk;m and Gk−1;mC1. Fig. 2. The graphs in Corollary 2.1. Proof. By the definitions of the Harary index and QG.vi/, we have X 1 X 1 H.G/ D C dG.vl; vk/ dG.vm; vi/ vl;vk2V .G−vi/ vm2V .G−vi/ X 1 D H.G − vi/ C dG.vm; vj/ C 1 vm2V .G−vi/ X dG.vm; vj/ D H.G − vi/ C 1 − dG.vm; vj/ C 1 vm2V .G−vi/ D − C − − H.G vi/ n 1 QG−vi .vj/; completing the proof of the lemma.
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