Open Journal of Discrete Mathematics, 2012, 2, 156159 http://dx.doi.org/10.4236/ojdm.2012.24031 Published Online October 2012 (http://www.SciRP.org/journal/ojdm) Cycles, the Degree Distance, and the Wiener Index* Daniel Gray1, Hua Wang2# 1Department of Mathematics, University of Florida, Gainesville, USA 2Department of Mathematical Sciences, Georgia Southern University, Statesboro, USA Email: dgray1@ufl.edu, #hwang@georgiasouthern.edu Received July 30, 2012; revised September 3, 2012; accepted September 11, 2012 ABSTRACT The degree distance of a graph G is , 11 1 2 nn iji ij DGdd L j , where and i d d are the degrees of vertices , and is the distance between them. The Wiener index is defined as , ij vv VG,ij L , 11 1 2 nn ij ij WG L . An elegant result (Gutman; Klein, Mihalić,, Plavšić and Trinajstić) is known regarding their correlation, that 4DTWT nn 1 for a tree T with n vertices. In this note, we extend this study for more general graphs that have frequent appearances in the study of these indices. In particular, we develop a formula regarding their correlation, with an error term that is presented with explicit formula as well as sharp bounds for unicyclic graphs and cacti with given parameters. Keywords: Degree Distance; Wiener Index; Cacti 1. Introduction The Wiener index of a graph is the sum of the distances between all pairs of vertices, denoted by G , 11 1 2 nn ij ij WG L where is the distance be ,ij L tween two vertices 12 ,:,, ij n vv VGvvv,. Such topological indices are defined as molecular descriptors that describe the structure/shape of molecules, helping to predict the activity and properties of molecules in com plex experiments. Among these indices, WG was introduced by and named after Wiener [1] as one of the most wellknown such concepts. Dobrynin and Kochetova [2] introduced the degree distance as a “degree analogue of the Wiener index”, de noted by , 11 , 11 1 2 nn ijij ij nn iij ij DGdd L dL where is the degree of the vertex . i The properties of and of various types of graphs are vigorously studied, see for instance [39] and the references there for such results on trees, unicyclic and bicyclic graphs, and cacti. In many cases their extremal values are achieved by the same structures. Then it is natural to consider the correlation between di v WG DG WG and DG . An elegant result for trees was achieved in as early as 1992, that Theorem 1.1. ([10,11]) 41DTWT nn for a tree T with n vertices. We generalize Theorem 1.1 to unicyclic graphs and cacti in general. In Section 2, we provide an observation where an “error term” is defined to simplify notations. With this observation, we provide the relation between DT and WT for unicyclic graphs and cacti in Section 3. 2. The “Error Term” and a Simple Observation Theorem 1.1 also follows from the fact that 0eT (for a tree ) in the following: T Lemma 2.1. Let an “error term”be 2 1DGnEG nn :4G WeG . Then , 11 11 2 ii j i ij ji eGdLd WG . (1) nn *This work was partially supported by a grant from the Simons Founda tion (#245307 to Hua Wang). #Corresponding author. Proof. From the definition we have C opyright © 2012 SciRes. OJDM
D. GRAY, H. WANG 157 ,, 11 , 11 2 , 11 =11 112 2 11 22 1. nn iijiji ij nn iij ij nn iiji ij ji DGdLL d dL WG nE Gn dL d WGnEG nn 1 To understand (1), note that for any two nonadjacent vertices and in , the sum y VG , 11 , 11 1 11 11 nn iiji ij ji nn n iij ij i ji dL d dL i d y en (2) counts the distance between x and y once when vi is a neighbor of x on the shortest path between x and y and j, and once when i is a neighbor of y on the shortest path between x and y and j. When i is summed over all i we get twice the number of edges, which double counts all pairs of vertices distance 1 apart. v vvxd Throughout this note, we will focus on the distances that are counted more than twice in (2), the sum of which gives us . Theorem 1.1 follows from Lemma 2.1 through the fact that all distances are counted exactly twice in a tree. Also, note that the sum of distances of pairs of adjacent vertices are counted exactly twice by 1. Hence, we only need to consider the distances between nonadjacent vertices. eG n i id 3. Unicyclic Graphs and Cacti In this section, we extend this result to unicyclic graphs and cacti. Although the unicyclic graphs can be con sidered as a special case of cacti, it is convenient to illustrate the proof by studying a unicyclic graph first. The “error term” eG ables us to present the general formula in a “neat” manner, then we focus on its explicit form and bounds. A cactus is a connected graph where any two cycles share at most one vertex. Trees, cycles, and unicyclic graphs are all special cases of cacti. Let G be the number of cycles in , then in cycles and unicyclic graphs. G1s 3.1. Unicyclic Graphs Theorem 3.1. for a unicyclic graph on vertices. 41DGWG nneG Gn 2 21 11 nn e nn G (3) where is the length of the unique cycle. Proof. The formula for follows immediately since DG ==EGGV n . 1) We first establish a formula for eG . Let 12 ,,, xx be the vertices on the cycle labeled in a clockwise fashion and let 12 ,,, XX be the corresponding components after removing all edges on the cycle. Note that k is a tree for 1,k2, , . First notice that the distances between any two vertices in k are counted exactly twice since k is a tree, for 1, 2,,k . Suppose that ik and vX l vX , and 1<kl . For ik vx , the shortest paths between v and all but one neighbors of i v must contain the shortest path from i to v v. Then the distance between v and all neighbors of i not on the path between v v and is counted exactly twice. i v If , let =vx ik be even (the case for odd is si milar). If 2 2 lk or 2 2 lk , then the short est paths between v and all but one neighbors of k must contain the shortest path from k to v. Then the distance between v and any vertex adjacent to k not on the shortest path between v and k is counted exactly twice. When 1 2 lk , the distance between v and any neighbor of k in VX are counted twice as above. k However, the distance between 1k and v is counted as 1,1,, , 2 kjkl lj lj Lx vLxxLxv Lxv ,Lvu distanc where is the distance between and . Note v u that thise is also counted for the case 1 2 lk , hence over counted once for every l vVX eG. Then the total contribution to as l ranges from 1 to is 1 1 , 2 ,. 2 l lvV X l l lvVX l Lxv nLxv When 2 lk , the distance between 1k or 1k is miscounted once as 1, 2l Lxv . and l vVX Copyright © 2012 SciRes. OJDM
D. GRAY, H. WANG Copyright © 2012 SciRes. OJDM 158 Then the contribution to is given by 3.2. eG G for General Cacti eG 1 1 1,. 2 l l lvVX l nLxv Then 1 Here xv is often referred to as the of Let be a cactus with cycles and r edges not on any cycle. Label the cycles c for 1, 2,, , let be the length of c and l be a vertex on c with component l (the component containing l after the removal of the edges on c ) for 1,2,,l . Define the distance function of l by 1 , 2l lvV X Lxv ,x ll vX l DL v . 1 1 2, 21. l lvVX l l l eGLx vn Dn Since 1EGn s , we immediately have 421DGWG nnseG . As was the case for the unicyclic graph, for every cycle in we have a contribution of G , ll vVX l DL distance function 1 2, lj lvVX lLx vn 1 to eG. Then an l in l . 2) Nalue eG. It is known that, ext we analyze v f sinimized by the ce e th with given number overtice, explicit formula is given by l D is m by onnter of a star and maximized e end of a path. Hence 1 11. ll ll VXDVX VX 11 11 21 22 s l l s l l eGD n Dnnrs 1. (4) 2 The sharp bounds of are given by eG 1 11 . ll l n VX VXn The sharp lower bound can be achieved by any cycle with pendant edges attached, the sharp upper bound can be With (4), we claim that, with given and cycle lengths, the starshaped cactus (a cactus that has only one cut vertex as its center) minimizes . ,,nsr eG 1 211 l l VX eG Claim 3.1. For any cactus with order n, s cycles, edges and any vertex , there exists a star shaped cactus with the same parameters and cycle lengths with center , such that G wVr G Su ,, vVS vVG LvuLvw . achieved by appending a path to a cycle, proving (3). In the case of a cycle, 0 l D and n , we have a simple corollary as follows. One can easily see the idea from Figure 1, where either operation from to G will reduce , vVGLvw Corollary 3.2. 44 nnn DC WC WC unless we have a starshaped cactus. 1nn eC cy rtices, where n eC nn Note that Corollary 3.2 can be directly verifi n 1 for a . ini Then eG is minimized when is a starshaped cactus. Consequently for all but one , for any given G =0 l D l . cle n C on n ve ed from the deftions, ,, 11 11 , 11 11 1 221 1 22 22 s l l s eGDnn rs rsnnrs 11 22 2 nn nn DGdd LL 2 24 . ij ijij ij ij nn ij ij LWG 21. Figure 1. Two operations that decreases e(G).
D. GRAY, H. WANG 159 Remark 3.3. If one does not specify the cycle lengths, then bounds similar to the unicyclic case can be achieved. Also, it is interesting to see that is minimized by a starshaped cactus, which miumber of graphi cal indices including ([ cacti. REFERENCES [1] H. Wiener, “Structural Determination of Paraffin Boiling Points,” Journal of the American C 69, No. 1, 1947, pp. 1720. doi:10.10 Vol. 63, No. 1, 2010, pp. 101112. [6] M. Fischermann, A. Hoffmann, D. Rautenbach, L. A. Székely and L. Volkmann, “Wiener Index versus Maxi mum Degree in Trees,” Discrete Applied Mathematics, Vol. 122, No. 13, 2002, pp. 127137. doi:10.1016/S0166218X(01)003572 eG nimizes a n 7]) among WG [7] H. Liu and M. Lu, “A Unified Approach to Cacti for Dif ferent Indices,” Match, Vol. 58, No. 1, 2007, pp. 183194. [8] A. I. Tomescu, “Properties of Connected Graphs Having tance,” Discrete Applied Mathe , No. 9, 2009, pp. 27452748. hemical Society, Vol. 21/ja01193a005 [2] on an 008 A. A. Dobrynin and A. A. Kochetova, “Degree Distance of a Graph: A Degree Analogue of the Wiener Index,” Journal of Chemical Informatid Computer Sciences, Vol. 34, No. 5, 1994, pp. 10821086. doi:10.1021/ci00021a man, S. “On the Degree Distance of a Graph,” Discrete Applie Mathematics, p. 27732777. [3] P. Dankelmann, I. Gut Mukwembi and H. C. Swart, d Vol. 157, No. 13, 2009, p doi:10.1016/j.dam.2009.04.006 [4] A. A. Dobrynin, R. Entringer and I. Gutman, “Wiener Index of Trees: Theory and Applications,” Acta Appli candae Mathematicae, Vol. 66, No. 3, 2001, pp. 211249. doi:10.1023/A:1010767517079 [5] Z. Du and B. Zhou, “Minimum Wiener Indices of Trees and Unicyclic Graphs of Given Matching Number,” Match, Minimum Degree Dis matics, Vol. 309 doi:10.1016/j.disc.2008.06.031 [9] A. I. Tomescu, “Unicyclic and Bicyclic Graphs Having Minimum Degree Distance,” Discrete Applied Mathe matics, Vol. 156, No. 2, 2008, pp. 125130. doi:10.1016/j.dam.2007.09.010 [10] I. Gutman, “Selected Properties of the Schultz Molecular Topological Index,” Journal of Chemical Information and Computer Sciences, Vol. 34, No. 5, 1994, pp. 10871089. doi:10.1021/ci00021a009 [11] D. J. Klein, Z. Mihalic′, D. Plavšic′ and N. Trinajstic′, “Molecular Topological Index: A Relation with the Wie ner Index,” Journal of Chemical Information a puter Sciences, Vol. 32, No. 4, 1 nd Com 992, pp. 304305. Copyright © 2012 SciRes. OJDM
