Wiener Odd and Even Indices on BC-Trees

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1 Wener Odd and Even Indces on BC-Trees Yu Yang, Lang Zhou, Hongbo Lu,3, Ajth Abraham 3, School of Informaton, Dalan Martme Unversty, Dalan 60, Chna School of Computer, Dalan Natonaltes Unversty, Dalan 6600, Chna 3 Machne Intellgence Research Labs (MIR Labs), WA, USA 3 ITInnovatons, VSB-Techncal Unversty of Ostrava, Czech republc yangyugdzs@gmal.com, lhb@dlmu.edu.cn, ajth.abraham@eee.org Abstract Correspondng to the concepts of Wener ndex and dstance of the vertex, n ths paper, we present the concepts of Wener odd (even) ndex of G as sum of the dstances between all pars of vertces of G satsfyng the dstances are all odd (even) and denote them W odd (G) and W odd (G) respectvely. Based on the concepts of the two ndces, we prove theoretcally that Wener odd ndex s not more than ts even ndex for general BC-Trees. Closed formulae of the two ndces are also provded for path BC-tree, star, k-extendng star tree and caterpllar BC-tree. Meanwhle, the extreme values of W odd (T ) of n vertces BC-trees are characterzed as well. Keywords-Wener odd (even) ndex; odd (even) dstance of the vertex; BC-tree; k-extendng star; caterpllar BC-tree; I. INTRODUCTION All graphs G = (V (G), E(G)) n ths paper wll be a fnte, undrected and connected, V (G) and E(G) denote the vertex set and the edge set of G, respectvely. Let d G (u, v) (or smply d(u, v) when no confuson arses) denote the dstance between the vertces u and v n G. For a vertex v of G, defne the dstance of the vertex as the sum of dstances from v to all other vertces. g G (v) = d G (v, u), u V (G) defne the odd(even) dstance of the vertex v as the sum of dstances from v to all other vertces of G satsfyng the dstances are all odd(even)..e. g odd,g (v) = d G (v, u), and g even,g (v) = Let W (G) = u V (G) d G (v,u) (mod ) u V (G) d G (v,u) 0(mod ) v V (G) d G (v, u). g G (v) denote the Wener ndex of G, whch s the sum of dstances for all unordered pars of vertces. Let W odd (G) = g odd,g (v)(w even (G) = v V (G) g even,g (v)) denote the Wener odd(even) ndex of v V (G) G, whch s the sum of dstances between all unordered pars of vertces satsfyng the dstances are all odd(even). Obvously, we have g G (v) = g odd,g (v) g even,g (v), W (G) = W odd (G) W even (G) A tree T = (V, E) s a connected, acyclc graph. A vertex of degree one wll be called a pendent vertex or leaf of T. A tree on n vertces has at least and at most n pendent vertces. The (unque) n-vertex trees wth and n pendent vertces are called the path and star, respectvely, and are denoted by P n and K,n, respectvely. A tree s called a BC-tree (the block-cutpont-tree or the bcolorable tree) f the dstance between any two leaves s even see Harary and Plummer [], []. BC-tree have many specal propertes and applcatons n graph theory, chemstry, computer and bran scence, see [3], [], [5], [6], [7]. Defne k-extendng star tree the tree constructed by addng n paths wth length k to each of the n pendant vertces of the star K,n denoted by K k,n, the center vertex s denoted as c, for any vertex u V (K k,n ), Let the dstance d K k,n (u, c) between u and c the heght of u. By ths defnton, K k,n s the star K,n when k =. A caterpllar tree s a tree, whch has a path, such that every vertex not on the path s adjacent to some vertex on the path. A caterpllar BC-tree s both a caterpllar tree and a BC-tree. P n s also a BC-tree, when n s odd. For standard notatons n graph theory, [8], [9] may be consulted. The rest of the paper s organzed as follows. Secton II descrbes related works on Wener ndex and BC-trees. In Secton III, we study the relatonshp between Wener odd ndex W odd (G) and Wener even ndex W odd (G) of general BC-tree, path BC-tree, star, k-extendng star tree, caterpllar BC-tree, and we also gve out the maxmum and mnmum value of W odd (T ) and the extremal trees attanng these values as well. Fnally the conclusons and some open problems are presented n Secton IV. II. RELATED WORKS The Wener ndex was frst developed by Wener [0] n 97. And the graphcal nvarant W (G) has been studed n [], [], [3], [] under dfferent names such as dstance, transmsson, total status and sum of all dstances. Ths

2 concept has been one of the most wdely used descrptors n quanttatve structure actvty relatonshps, as the Wener ndex has been shown to have a strong correlaton wth the chemcal propertes of a chemcal compound [5]. The Wener ndex and the average dstance rank among those graph-theoretcal parameters that are of most nterest n other felds. Dobrynn [] provded a very comprehensve survey about Wener ndex. Chemcal structures of organc compounds are characterzed numercally by a varety of structural descrptors, one of the earlest and most wdely used beng the Wener ndex W, derved from the nteratomc dstances n a molecular graph. Extensve use of such structural descrptors or topologcal ndces has been made n drug desgn, screenng of chemcal databases, and smlarty and dversty assessment. A new set of topologcal ndces s ntroduced representng a parttonng of the Wener ndex based on counts of even and odd molecular graph dstances by Ivancuc [6]. These new ndces are further generalzed by weghtng exponents whch can be optmzed durng the quanttatve structure-actvty/- property relatonshp (QSAR/QSPR) modelng process. In [6], Ivancuc also tested these novel topologcal ndces n QSPR models for the bolng temperature, molar heat capacty, standard Gbbs energy of formaton, vaporzaton enthalpy, refractve ndex, and densty of alkanes, and Ivancuc [6] conluded that n many cases, the even/odd dstance ndces proposed here gve notably mproved correlatons. Namely, lke the Wener ndex, the Wener odd and even ndex of graph may also be meanngful topologcal ndex, whch can be used n mathematcs, chemstry, bonformatcs and bran scence. Hence, the study of Wener odd and even ndex of graph s of great sgnfcance. III. WIENER ODD (EVEN) INDEX ON BC-TREES A. Wener odd (even) ndex on general BC-trees Snce the concepton of the block-cutpont-tree (.e. BCtree) was proposed, t has been drawng researchers attenton and concern worldwde, not only n the feld of mathematcs and computer scence, but also n chemstry and bran scence. See Barefoot [5], Mkrtchyan [7], Msołek and Chen [8], Paton [9], Gagarn and Labelle [0], Nakayama and Fujwara [], Yang and Wang [] It s well known that the Wener ndex among trees on n vertces s mnmzed by the star K,n and s maxmzed by the n-vertex path P n, see Entrnger et al. [3], or Lovász []. Snce BC-tree s a tree wth specal nature, next, we wll examne the the Wener odd (even) ndex on BC-tree. Lemma : [3] The Wener ndex of n-vertex path P n s (n ), more than any other tree on n vertces. The Wener ndex of star K,n s (n 3 n)/6, fewer than any other tree on n vertces. Theorem 3.: Let T be a BC-tree on n vertces, then W odd (T ) W even (T ). Proof: For any BC-tree T, we can always fnd a vertex v, such that each two trees after splttng at v are also BCtrees, and one of them s a star. For llustraton see Fg.. Let T be a star on p (p ) vertces, for brevty, set S = {w w V (T )\v d T (w, v) 0(mod )}, S = {w w V (T ) d T (w, v) (mod )}, R = {w w V (T )\v d T (w, v) 0(mod )}, R = {w w V (T ) d T (w, v) (mod )}, N = S, N = S, M = R, M = R, g even,t (v) = w S d T (w, v), g odd,t (v) = w S d T (w, v) g even,t (v) = T v BC tree T w R d T (w, v), g odd,t (v) = u T Splttng T at v BC tree T Fgure. Splt of BC-tree T. T w R d T (w, v) v u BC tree T Snce T s a BC-tree, By the defnton of BC-tree, t s not dffcult to obtan N N, g odd,t (v) g even,t (v). () It s clear that the theorem holds for n = 3,, 5, suppose the theorem holds for BC-trees of orders < n, then, we categorse the unordered pars of vertces of T satsfyng the dstances are all odd as follow: (){(p, q) p V (T ), q V (T ) d T (p, q) (mod )}; (){(p, q) p V (T ), q V (T ) d T (p, q) (mod )}; (3){(p, q) p R, q S }; (){(p, q) p R, q S }. Smlarly, the unordered pars of vertces of T satsfyng the dstances are all even are as follow: (5){(p, q) p V (T ), q V (T ) d T (p, q) 0(mod )}; (6){(p, q) p V (T ), q V (T ) d T (p, q) 0(mod )}; (7){(p, q) p R, q S }; (8){(p, q) p R, q S }. Wth the above notatons, t s easy to know that M = p, M =. By analyzng, the sum of the dstances of the unordered pars of vertces of cases (3)() and cases (7)(8) s p g odd,t (v) pn g even,t (v) N and p g even,t (v) pn g odd,t (v) N respectvely. T

3 Obvously, cases () and () are W odd (T ) and W odd (T ); cases (5) and (6) are W even (T ) and W even (T ). Therefore, W odd (T ) W even (T ) = W odd (T ) W even (T ) W odd (T ) W even (T ) (p )(g odd,t (v) g even,t (v)) (p )(N N ) () By nducton, W odd (T ) W even (T ), W odd (T ) W even (T ), snce p, combnng (), we obtan W odd (T ) W even (T ). The theorem thus holds. Theorem 3.: The Wener odd ndex of star K,n s n, fewer than any other BC-trees on n vertces and the maxmum value of Wener odd ndex s (n 3 n)/. Proof: Let T be an arbtrary n vertces BC-tree, It s easy to see that W odd (T ) n, and t s not hard to observe that W odd (K,n ) = n, therefore, star K,n mnmzes the Wener odd ndex. Next, we prove any other BC-tree on n vertces can not attan ths value. Suppose l s a leaf of T, v s the neghbor vertex of l, f the dstances between all other vertces and v s, then they are neghbors of v, otherwse, there wll exst a vertex w, such that d T (l, w) = 3, then W odd (T ) > n. Hence T s star. By Theorem 3. and Lemma, we have W odd (T ) W (T )/ = (n 3 n)/. By smple calculaton, we have W odd (P n ) = W even (P n ) = (n 3 n)/ for n s odd, but t sn t the unque BC-tree that can attan ths maxmum value. Székely and Wang [5] studed the problem of enumeratng subtrees of trees. They proved the followng results: ( Lemma : (Székely and Wang [5]) The path P n has n ) subtrees, fewer than any other trees of n vertces. The star K,n has n n subtrees, more than any other trees of n vertces. In [], we studed the BC-subtrees of K,n and P n. And we have the followng theorems. Lemma 3: []The star K,n has n n BCsubtrees, more than any other trees of n vertces. Lemma : []The number of BC-subtrees of path P n s { n(n )/ n 0(mod ), η BC (P n ) = (n ) / n (mod ). fewer than any other n-vertces tree. We know that star K,n s a BC-tree, and P n s also a BC-tree when n s odd. By Theorem 3., Lemma 3, Lemma we have K,n mnmzes the Wener odd ndex and maxmzes the BC-subtrees among BC-trees on n vertces; P n maxmzes the Wener odd ndex and mnmzes the BCsubtrees among BC-trees on n(n s odd) vertces. We see here an amazng and not yet understood relatonshp between the Wener odd ndex and the number of BC-subtrees whch s wthn certan classes of BC-trees of a fxed parameter, the smaller the number of BC-subtrees s, the bgger the Wener odd ndex s. Unfortunately ths relatonshp does not extend as expected. See Fg.. T 0 and T 0 are BC-trees on vertces. Smple calculatons show that W odd (T 0 ) = 69, W odd (T 0) = 73, η BC (T 0 ) = 83, η BC (T 0) = 5(where η BC (T 0 ) denotes the BC-subtrees number of T ), we have W odd (T 0 ) < W odd (T 0), though η BC (T 0 ) < η BC (T 0). BC-tree T0 BC-tree Fgure. The counter-example caterpllar tree T B. Wener odd (even) ndex on k-extendng star trees Snce k-extendng star tree wth branches s a path BC-tree, by Theorem 3., ts Wener even ndex equals ts Wener odd ndex. Thus, we only consder n. Theorem 3.3: Let K,n k be a k-extendng star tree wth n (n ) branches, then, W even (K,n ) k equals W odd (K,n ) k (resp. W odd (K,n ) k (k )(n )(n 3)/) for n s even(resp. odd). Proof: We dscuss t for k s even and odd respectvely. () For k s even, usng the splttng method and denotatons as n Theorem 3., and for the sake of convenence, for branch b ( =,,..., n ), label the vertces of b as ths: b = v 0 v v... v k v k v k where v 0 s the center vertex c and v k s the leaf vertex, then we splt K,n k at vertex v k j (j =,, 6,..., k, k) sequentally each tme, T and T are as n Fg. 3(a) after the frst splttng. Smple calculatons show that W odd (T ) = W even (T ), P =, and N = N, hence, by (), W odd (T ) W even (T ) = W odd (T ) W even (T ). Contnue the splttng process untl the whole branch b s deleted, denote the tree at ths tme as T 0, then W odd (T ) W even (T ) = W odd (T 0 ) W even (T 0 ) contnue deletng ths way untl T becomes a k-extendng star tree wth branches, we denote t as T 0, then we have W odd (T ) W even (T ) = W odd (T 0) W even (T 0). Snce W odd (T 0) = W even (T 0), therefore, W odd (T ) = W even (T ). () For k s odd, as the dscusson for n s even, smlarly, after the frst splttng, we have W odd (T ) W even (T ) = W odd (T ) W even (T ) (n 3) by (). Proceedng the splttng untl T s spltted to a tree wth only one leaf n heght and the other n leaves are n heght k, see Fg. T 0

4 the length s k T c( v 0) v v v k v k v( v k ) T whch we call k-tree and lobster graph. Caterpllar trees have been used n chemcal graph theory to represent the structure of benzenod hydrocarbon molecules and receved more and more attenton by the researchers. Combnng the specal property of BC-trees, next, we examne the caterpllar BC-tree. Consder the caterpllar tree T wth dameter l (l >= )(l s even by defnton)n Fg., x and y are leaves on the dameter. After the deleton of all the edges of P T (x, y) from T, some connected components wll reman. Let T denote the star on n vertces wth center vertex v, for =,,..., (l ). (a) The splttng llustraton of K,n k even) (k s n n n n n l / 3 ( l )/ x v w v w v 3 v( l v w w 3 ( l )/ )/ w( l )/ T T T 3... T( l )/ T y the length s k c( v 0) v Fgure. The caterpllar BC-tree T wth dameter l. BC-tree T (b) The splttng llustraton of K,n k (k s odd) Fgure 3. The splttng llustraton of K k,n Theorem 3.: Let T be the caterpllar BC-tree descrbed above. Then, W even (T ) > W odd (T ) and W even (T ) = (l )/ W odd (T ) n (n ) (n ) (l )/ n ( j= j= (j )(n j ) (l ) /). n j 3(b), and we denote at ths tme as T, then we have W odd (T ) W even (T ) = W odd (T ) W even (T ) (k )(n 3)/. Next, choose another branch, go on splttng, untl T s splt to a star K,n, then we have W odd (T ) W even (T ) = W odd (K,n ) W even (K,n ) (n )(k )(n 3)/. By Lemma and Theorem 3., we have W odd (K,n ) W even (K,n ) = (n )(n 3). Therefore W even (T ) = W odd (T ) (k )(n )(n 3)/. The theorem holds. C. Wener odd (even) ndex on caterpllar BC-trees Caterpllars were frst studed n a seres of papers by Harary and Schwenk [6], [7]. And they have many characterzatons and generalzatons such as a chordal graph wth exactly n k maxmal clques, each contanng k vertces Proof: We splt the T as descrbed n Theorem 3. at vertex w ( =,,..., (l )/) sequentally, then two parts obtaned each tme were denoted as T and T, where T s a star on n 3 vertces and T s a caterpllar BCtree, and t s easy to observe that T = K,n( =,,..., (l )/), T ( =,,..., (l )/) s a caterpllar BC-tree. After splttng (l )/ tmes, the caterpllar BCtree T was spltted up. For the sake of brevty, we gve some denotatons. S, = {w w V (T )\w d T (w, w ) 0(mod )}, S, = {w w V (T ) d T (w, w ) (mod )}, N, = S,, g even,t (w ) = w S, d T (w, w ), N, = S,, g odd,t (w ) = d T (w, w ). w S, Meanwhle, we defne p = n ( =,,..., (l ) ).

5 After splttng T at w, Smple calculatons show that By (), we have N, = n j, N, =, j= g even,t (w ) = (j )(n j ), j= g odd,t (w ) = (l ) /, p = n, W odd (T ) W even (T ) = n (n ). W odd (T ) W even (T ) = W odd (T ) W even (T ) n (n ) n ((l ) / (j )(n j )) (n ) n j. j= j= Proceedng ths way, after splttng (l )/ tmes, we have W odd (T ) W even (T ) = W odd (K,n ) W even (K,n ) (l )/ (l )/ (l )/ (W odd (T ) W even (T )) (p )(g odd,t (w ) g even,t (w )) (p )(N, N, ). Proof: We splt T ϕ and T φ at vertex w ( =,,..., l ), w ( =,,..., l ) sequentally, then, after some algebrac operatons, When l = k, W odd (T ϕ ) = l/ j= l 8j l lj 8j W odd (T φ ) = l/ j= l 8j l lj 8j (n When l = k, W odd (T ϕ ) = (l )/ j= l 8j l lj 8j l l (n (l)/ 8 ) W odd (T φ ) = (l )/ j= (n j n l ) j j n l ) j (n j n l ) j l 8j l lj 8j (n j n l l l (n (l)/ 8 ) j ).e. W odd (T ) W even (T ) = W odd (K,n ) (l )/ W even (K,n ) n (n ) (l )/ (l )/ n ((l ) / (n ) j= n j. j= (j )(n j )) Snce W odd (K,n ) W even (K,n ) = n (n ), the theorem thus holds. Let T ϕ, T φ be caterpllar BC-trees wth the structure as depcted n Fg. wth n ( =,,..., l ), w ( =,,..., l ) and n ( =,,..., l ), w ( =,,..., l ) respectvely. Theorem 3.5: Let T ϕ, T φ be descrbed above satsfng the followng condton: n n l = n n l ( =,,..., l ), then, W odd(t ϕ ) = W odd (T φ ). In ether case, we know that f n n l = n n ( =,,..., l l ), then, W odd(t ϕ ) = W odd (T φ ). IV. CONCLUSIONS AND FURTHER WORKS In ths paper, we presented the concepts of odd (even) dstance of the vertex v as the sum of dstances from v to all other vertces of G satsfyng the dstances are all odd (even). We llustrated Wener odd (even) ndex of G as the sum of the dstances between all pars of vertces of G satsfyng the dstances are all odd (even), whch are denoted as W odd (G) (W even (G)) respectvely. And we proved that the Wener odd ndex s not more than ts even ndex for general BC-Trees. For k-extendng star trees and caterpllar BC-trees, we gave out the equaltes of ther Wener even ndex and odd ndex. We also gave out the maxmum value (n 3 n)/ and mnmum value n of W odd (T ) on n vertces BC-trees and the extremal BC-trees attanng these values as well. As the wdely researched Wener ndex, we can also do some researches on odd (even) dstance of vertex and

6 Wener odd (even) ndex on specal trees and general trees correspondngly and comprehensvely; Obvously, Wener odd and even ndex can also be regarded as topologcal ndex. Hence another nterestng drecton s to explore the role of these ndex n graph theory, chemstry and bran networks. ACKNOWLEDGMENT The authors would lke to thank Dr. Hua Wang for hs scentfc collaboraton n ths research work. Ths work s supported partly by the Natonal Natural Scence Foundaton of Chna (Grant No ) and the Program for New Century Excellent Talents n Unversty (Grant No. NCET--086). Ajth Abraham acknowledges the support of the ITInnovatons Centre of Excellence project, reg. no. CZ..05/..00/ funded by Structural Funds of the European Unon and state budget of the Czech Republc. REFERENCES [] F. Harary and G. Prns, The block-cutpont-tree of a graph, Publcatones Mathematcae - Debrecen, vol. 3, pp , 966. [] F. Harary and M. Plummer, On the core of a graph, Proceedngs London Mathematcal Socety, vol. 7, pp. 9 57, 967. [3] R. Johannesson and K. Zgangrov, On the dstrbuton of the number of computatons n any fnte number of subtrees for the stack algorthm (corresp.), IEEE Transactons on Informaton Theory, vol. 3, no., pp. 00 0, 985. [] J. Barnard, A comparson of dfferent approaches to markush structure handlng, Journal of Chemcal Informaton and Computer Scences, vol. 3, no., pp. 6 68, 99. [5] C. Barefoot, Block-cutvertex trees and block-cutvertex parttons, Dscrete Mathematcs, vol. 56, no., pp. 35 5, 00. [6] A. Symeonds and I. Tolls, Vsualzaton of bologcal nformaton wth crcular drawngs, Bologcal and Medcal Data Analyss, Lecture Notes n Computer Scence, vol. 3337, pp , 00. [7] F. De Vco Fallan, L. Astolf, F. Cncott, D. Matta, A. Tocc, S. Salnar, M. Marcan, H. Wtte, A. Colosmo, and F. Bablon, Bran network analyss from hgh-resoluton eeg recordngs by the applcaton of theoretcal graph ndexes, IEEE Transactons on Neural Systems and Rehabltaton Engneerng, vol. 6, no. 5, pp. 5, 008. [8] J. A. Bondy, Graph Theory wth Applcatons. The Macmllan Press Ltd, 979. [9] F. Harary, Graph Theory. Addson-Wesley, Readng, MA, 969. [0] H. Wener, Structural determnaton of paraffn bolng ponts, Journal of the Amercan Chemcal Socety, vol. 69, no., pp. 7 0, 97. [] A. Dobrynn, R. Entrnger, and I. Gutman, Wener ndex of trees: theory and applcatons, Acta Applcandae Mathematcae, vol. 66, no. 3, pp. 9, 00. [] M. Fschermann, A. Hoffmann, D. Rautenbach, L. Székely, and L. Volkmann, Wener ndex versus maxmum degree n trees, Dscrete Appled Mathematcs, vol., no., pp. 7 37, 00. [3] F. Jelen and E. Tresch, Superdomnance order and dstance of trees wth bounded maxmum degree, Dscrete Appled Mathematcs, vol. 5, no., pp. 5 33, 003. [] S. Klavžar and I. Gutman, Wener number of vertexweghted graphs and a chemcal applcaton, Dscrete Appled Mathematcs, vol. 80, no., pp. 73 8, 997. [5] I. Gutman and T. Kortvelyes, Wener ndces and molecular surfaces, Zetschrft für Naturforschung. A, A Journal of physcal scences, vol. 50, no. 7, pp , 995. [6] O. Ivancuc, T. Ivancuc, D. Klen, W. Setz, and A. Balaban, Wener ndex extenson by countng even/odd graph dstances, Journal of Chemcal Informaton and Computer Scences, vol., no. 3, pp , 00. [7] V. Mkrtchyan, On trees wth a maxmum proper partal 0- colorng contanng a maxmum matchng, Dscrete Mathematcs, vol. 306, pp , 006. [8] E. Msolek and D. Z. Chen, Two flow network smplfcaton algorthms, Informaton Processng Letters, vol. 97, pp. 97 0, 006. [9] K. Paton, An algorthm for the blocks and cutnodes of a graph, Communcatons of The ACM, vol., pp , 97. [0] A. Gagarn and G. Labelle, Two-connected graphs wth prescrbed three-connected components, Advances n Appled Mathematcs, vol. 3, no., pp. 6 7, 009. [] T. Nakayama and Y. Fujwara, BCT representaton of chemcal structures, Journal of Chemcal Informaton and Computer Scences, vol. 0, no., pp. 3 8, 980. [] Y. Yang, D. Wang, H. Wang, and H. Lu, On BC-trees and BC-subtrees, arxv preprnt arxv:305.7, 03. [3] R. Entrnger, D. Jackson, and D. Snyder, Dstance n graphs, Czechoslovak Mathematcal Journal, vol. 6, no., pp , 976. [] L. Lovász, Combnatoral problems and exercses. Amercan Mathematcal Socety, 993. [5] L. Székely and H. Wang, On subtrees of trees, Advances n Appled Mathematcs, vol. 3, no., pp , 005. [6] F. Harary and A. Schwenk, The number of caterpllars, Dscrete Mathematcs, vol. 6, no., pp , 973. [7] S. El-Basl, Applcatons of caterpllar trees n chemstry and physcs, Journal of Mathematcal Chemstry, vol., no., pp. 53 7, 987.