Formulas for the Wiener number and the Hosoya-Wiener polynomial of edge and vertex
weighted graphs are given in terms of edge and path contributions. For a rooted tree, the Hosoya-
Wiener polynomial is expressed as a sum of vertex contributions. Finally, a recursive formula
for computing the Hosoya-Wiener polynomial of a weighted tree is given

Blaž Zmazek

Janez Žerovnik

English

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The Hosoya-Wiener Polynomial of Weighted TreesBla` Zmazeka,* and Janez @erovnikb,*aFME, University of Maribor, Smetanova 17, SI-2000 Maribor, SloveniabIMFM, Jadranska 19, SI-1000 Ljubljana, SloveniaRECEIVED NOVEMBER 14, 2006; REVISED DECEMBER 21, 2006; ACCEPTED JANUARY 11, 2007Formulas for the Wiener number and the Hosoya-Wiener polynomial of edge and vertexweighted graphs are given in terms of edge and path contributions. For a rooted tree, the Ho-soya-Wiener polynomial is expressed as a sum of vertex contributions. Finally, a recursive for-mula for computing the Hosoya-Wiener polynomial of a weighted tree is given.KeywordsHosoya-Wiener polynomialvertex weighted graphsweighted tree* Authors to whom correspondence should be addressed. (E-mail: B. Z., blaz.zmazek@uni-mb.si; J. @., janez.zerovnik@imfm.uni-lj.si)CROATICA CHEMICA ACTACCACAA 80 (1) 75¿80 (2007)ISSN-0011-1643CCA-3142Original Scientific PaperINTRODUCTIONIn this paper we consider the vertex and edge weightedgraphs where the weights are positive real numbers. Inmathematical chemisty, molecules are often modelled bygraphs. The sum of all distances is known as the Wienernumber.1 It was the first non-trivial topological index inchemistry, which was introduced by H. Wiener in 1947to study boiling points of paraffins (for historical data,see for instance Ref. 2). Since then, the Wiener number(also called the Wiener index) has been used to explainvarious chemical and physical properties of moleculesand to correlate the structure of molecules with their bio-logical activity. The research interest in the Wiener num-ber and related indices is still considerable.3,4The Hosoya-Wiener polynomial of a graph G is de-fined as:W(x) = W(G; x) = ∑ xd(u,v), (1)where the sum runs over all unordered pairs of verticesu, v ∈V (G). This definition, which is used for examplein Ref. 5, slightly differs from the definition used by Ho-soya6 (see also Ref. 7): ( )  ( ; ) ( , )W x W x x d u vu v= =≠∑G . (2)Obviously, W(x) = W(x) + |V(G)|.In his paper6 Hosoya used the name Wiener poly-nomial while some authors later used the name Hosoyapolynomial.8,9 We decided to use a compromise namehere, namely the Hosoya-Wiener polynomial.It is well known that the first derivative of the Hoso-ya-Wiener polynomial evaluated at x = 1 equals the Wienernumber (see for example Ref. 7). Higher derivatives ofthe Hosoya-Wiener polynomial have also been used asdescriptors.10,11In an arbitrary tree, every edge is a bridge, i.e., afterdeletion of an edge, the graph is no more connected. Thecontribution to the Wiener number of an edge was takento be the product of the numbers of vertices in the twoconnected components. This number also equals the num-ber of all shortest paths in the tree that go through theedge.12 Therefore the usual generalization of the Wienernumber on arbitrary graphs is defined to be the sum ofall distances in a graph. Recalling and generalizing theoriginal definition, edge contributions to the Wiener num-ber were studied in Refs. 13, 14, 15. Here we study edgeand path contributions to the Hosoya-Wiener polynomialof weighted graphs.We claim that the algorithm given in Ref. 16 can begeneralized to compute the Hosoya-Wiener polynomial.However, not only the sum of all distances, but the dis-tance distribution is nedeed for the polynomial. Hence, avector of distances has to be mantained and the complexi-ty of computation will necessarily include an additionalfactor related to the number of different distances that ap-pear in the graph. Here we discuss a generalization of treesand give a recursive formula for the Hosoya-Wiener poly-nomial of a rooted tree.PRELIMINARIESA weighted graph G = G(V,E,w,l) is a combinatorialobject consisting of an arbitrary set V = V (G) of vertices,a set E = E(G) of unordered pairs {x, y} = xy of distinctvertices of G called edges, and two weighting functions,w and l. w : V (G)  IR+ assigns positive real numbers(weights) to vertices and l : E(G)  IR+ assigns posi-tive real numbers (lengths) to edges. Without a loss ofgenerality, we will assume V (G) = {1, 2, 3, ..., n}.A simple path from u to v is a finite sequence of dis-tinct vertices P = x0, x1, ..., xl such that each pair xi–1, xiis connected by an edge and x0 = u and xl = v. The lengthof the path is the sum of the lengths of its edges, l(P) =il=∑ 1 l(xi–1 xi). For any pair of vertices u, v, we definethe distance d(u, v) to be the minimum of lengths overall paths between u and v. If there is no such path, wewrite d(u, v) = ∞.A graph G is connected if d(x, y) < ∞ for any pair ofvertices x, y. A tree is a connected graph with |E(G)| =|V(G)| – 1. A rooted tree is just a tree in which one ver-tex is called the root. For the usual graph theoretical ter-minology not defined here see, for instance, Ref. 17.HOSOYA-WIENER POLYNOMIALThe Hosoya-Wiener polynomial of a weighted graph Gis defined as:W(x) = W(G, l, w; x) = w u w v x d u vu v( ) ( ) ( , )≤∑ . (3)Recently, Klein, Do{li} and Bonchev18 generalized theHosoya-Wiener polynomial to vertex weighted graphs intwo ways that slightly differ from the one given here.Clearly, our definition is equivalent to the originaldefinition (1) if all vertex and edge weights are equal to1. Note that in this case we get W(G, 1, 1; x) = W(x) =u v V, ( )∈∑ G xd(u,v) like in Ref. 5.Remark. The generalization W(G, l, w; x) studiedhere may not be a polynomial if the edge weights areallowed to be arbitrary real numbers. Obviously, if na-tural numbers are used for edge weights, the functionW(G, l, w; x) is a polynomial. Hence, with appropriatescaling factor, one can always consider W(G, l, w; x) to bea polynomial, for any model using rational edge weights.The weighted Wiener number19 of a weighted graphG is:W = W(G, l, w) =w u w v d u vu v( ) ( ) ( , )<∑ = w u w v d u vu v( ) ( ) ( , )≤∑ . (4)This definition is clearly a generalization of the usual de-finition of the (unweighted) Wiener number. More pre-cisely, if all weights of vertices are 1 and all lengths ofedges are 1, then W(G, l, w) is the usual Wiener numberW(G).It seems that the weighted Wiener number has notbeen studied frequently in the literature. In Ref. 19, a li-near algorithm for computing the weighted Wiener num-ber is given. A definition, analogous to (4), was used inRefs. 20 and 21 for vertex weighted graphs. A differentdefinition, in which the »weights« of atoms are added tothe sum of distances, is given in Ref. 22. Weight distan-ces with the reverse of bond order are used in Ref. 23.Let us just mention perhaps the most interesting propertyof the Hosoya-Wiener polynomial, namely W'(G, l, w;x) = W(G, l, w).A REMARK ON EDGE AND PATHCONTRIBUTIONSWe first show how the weighted Wiener number of a ge-neral weighted graph can be seen as a sum of path andedge contributions.Lemma 1. – For a weighted graph G = G(V, E, w, l),W(G) = l( )ee uv=∑ ⋅ 1n a bw a w bP ea b* ( , )( ) ( ),* ∋∑ ,where P a b,* is a shortest path between a and b and n*(a;b) is the number of shortest paths with endpoints a and b.Proof. To see this, it is enough to sum up the contribu-tions of each edge to W(G) in two different ways. Eachpair a, b of vertices contributes w(a)w(b)d(a, b) to theWiener number. This contribution can be either regardedas a contribution of the pair a, b or it can be divided into76 B. ZMAZEK AND J. @EROVNIKCroat. Chem. Acta 80 (1) 75¿80 (2007)n*(a, b) path contributions, which can be further regardedas a sum of edge contributions along the path. Since d(a,b) is the sum of edge weights along a shortest path, thecontribution of edge:l(e)1n a b* ( , )w(a)w(b) .An edge contributes as many times as it appears onvarious shortest paths. Here the vertices are weighted, soone has to take into account the weights of both terminalvertices. Let us note in passing that the sum of contri-butions of the edge e:w*(e) :=1n a bw a w bP ea b* ( , )( ) ( ),* ∋∑ ,can be called the shortest path weight distribution, ge-neralizing the ideas of Refs. 13, 14, 15.Hence, one can sum up the lengths of all shortest paths(like in (4)), or, equivalently, sum up the contributions ofall edges.Recall that on a tree, there is a unique shortest pathbetween any pair of vertices. Hence n*(a, b) = 1 for alla, b.From Lemma 1 we read that the path contribution ofa shortest path P = P a b,* between vertices a and b is:1n a b* ( , )w(a)w(b) l( ),*ee Pa b∈∑ ,and the contribution of an edge e is:l(e) ⋅ 1n a bw a w bP ea b* ( , )( ) ( ),* ∋∑ .Note that the path contribution can be seen as a sum of(partial) edge contributions related to the path and theedge contribution can be seen as a sum of path contribu-tions related to the edge.However, the next lemma shows that for the Hosoya-Wiener polynomial, the path contributions can again beseen as sums of (partial) edge contributions related to thepath, but the situation is slightly different for the edgecontributions.Lemma 2. – For a weighted graph G,W(G, l, w; x) =1n a bw a w bPa b a b* ( , )( ) ( ),*∑∑<x ee Pa bl( ),*∈∏ ,where P a b,* is a shortest path between a and b and n*(a;b) is the number of shortest paths with endpoints a and b.Proof. To see this, it is enough to sum up the contribu-tions of each edge to H(G; l, w; x) in two different ways.Each pair a, b of vertices contributes w(a)w(b)xd(a, b) tothe Hosoya-Wiener polynomial. This contribution can beeither regarded as a contribution of the pair a, b or it canbe divided into n*(a, b) path contributions:x d(a,b)1n a bw a w b* ( , )( ) ( ) ,which can be further regarded as a product of edge con-tributions along the path.xd(a,b)1n a bw a w b* ( , )( ) ( ) =1n a bw a w b* ( , )( ) ( ) x ee Pa bl ( ),*∈∏ .Hence, one can sum up the contributions of all shortestpaths, but note that the edge contributions to the path con-tribution are multiplied, and not added. Representing theHosoya-Wiener in terms of edge contributions is hencesomewhat more complicated in the following sense. Foreach path crossing the edge, one needs to know the amountof traffic (in a possible natural model, the intensity of thetraffic corresponds to1n a bw a w b* ( , )( ) ( ), i.e., the numberof packets between a and b that use the path through e),but also the length of the paths. Edge contribution can begiven by:w(e) = xn a bw a w bd a bP ea b( , ),* * ( , )( ) ( )∋∑ 1 ,where, obviously, besides the amount of traffic some in-formation on the source and sink of the traffic is takeninto account.HOSOYA-WIENER POLYNOMIAL ON TREESAssume we are given a rooted tree T. For each vertex v∈T,let W xv ( ) denote the partial Hosoya-Wiener polynomialcorresponding to all distances from v to all other verticesin the subtree, rooted at v. By definition W xv ( ) = 0, if v isa leaf.Lemma 3. – Let v1, v2, ..., vk be the sons of a vertex v in arooted tree T with partial Hosoya-Wiener polynomialsW x W x W xv v vk  1 2( ), ( ), ... , ( ), respectively. Then, the partialHosoya-Wiener polynomial at vertex v can be computedas follows:W xv ( ) = w(v)W xw vw v xviiikv vi11( )( )( ) ( )+=∑ lProof. The path from v to vi contributes w(vi)xl(viv)w(v).The contribution of the path from v to w in the subtree,rooted at vi, can be written as:THE HOSOYA-WIENER POLYNOMIAL OF WEIGHTED TREES 77Croat. Chem. Acta 80 (1) 75¿80 (2007)w(w)x d(w,v)w(v) = w(w)x d(w,vi)+l(vi v)w(v) =w(w)x d(w,vi)xl(vi v)w(v) =w(w)x d(w,vi)w(vi)1w vi( )xl(vi v)w(v)Hence, the sum over all vertices w from the subtreerooted at vi and the sum over all sons vi of v gives theresult claimed:W x w v x w wv d w vw Tv ( ) ( ) ( )( , )= =∈∑w v w w x w vw vxw Td w viiv vikvi i( ) ( ) ( )( )( , ) ( )∈ =∑ ∑ =11lw vw vx w w x w viv vikd w viw Ti ivi( )( )( ) ( )( ) ( , )11l= ∈∑ ∑ =w vw vx W x w viv vv iikii( )( )( ( ) ( ) )( )1 21l +=∑ =w vW xw vw v xviiv vikii( )( )( )( ) ( )+=∑ l1.For each vertex v ∈T, let Wv (x) denote the contri-bution of the vertex v to the Hosoya-Wiener polynomial.This contribution corresponds to all shortest paths in thesubtree, rooted at vertex v, which meet v.On the other hand, each path P in a rooted tree meetsa unique vertex v at a minimal distance from the root.The contribution of the path P is thus counted in the Wv (x),and at no other vertex. Furthermore note that each pathmeets at least one vertex that is not a leaf.Lemma 4. – Let v1, v2, ..., vk be the sons of a vertex v in arooted tree T with partial Hosoya-Wiener polynomialsW x W x W xv v vk  1 2( ), ( ), ... , ( ), respectively. Then, the con-tribution Wv (x) of the vertex v to the Hosoya-Wienerpolynomial is:Wv(x) =121 12w vW x w vw vW x w vviv ii( )( ) ( )( )( ) ( ) +  + + =∑ik1⋅ xw vW x w v xv viv iv viiil l( ) ( )( )( ) ( )− + ⋅1   +=∑221w vik( )Proof. Any shortest path that meets vertex v consists ofone or can be built from two shortest paths from v (i.e.,either a path of the form v ... x or a concatenation of twopaths of the form x ... v ... y).The contribution of vertex v to the Hosoya-Wienerpolynomial is therefore the sum of:xw vw vW xv vivi il( ) ( )( )( )for all neighbors vi of v andx xw v w vW x W xv v v vi jv viji jl l( ) ( )( ) ( )( ) ( )1 over all pairs of neighbors of v. For brevity, let us denote:Ci =1w vW x w v xiv iv vii( )( ) ( ) ( ) + ⋅l .Note that Lemma 3 in new notation reads W xv ( ) =w(v) ⋅ Ciik=∑ 1 . Obviously,1212w vW x w v w v Cv iik( )( ) ( ) ( ) +  = + ==∑w v w v C C Ci i jjkikik( ) ( )2111+ + ====∑∑∑w v( )2 + w(v) Ciik=∑1+ C Ci ji j<∑ + Ciik21=∑ + C Ci ji j>∑and because C Ci ji j<∑ = C Ci ji j>∑ , we haveC Ci ji j<∑ =121212 21w vW x w v w v C C w vv iikiik( )( ) ( ) ( ) ( ) +  − − −= =∑ ∑.The shortest paths in the subtree rooted at vertex vthat meet v include paths with none, one or both endpointsequal vertex v. (The last case corresponds to a path ofdistance 0.) Hence:Wv(x) = w(v) Ciik=∑1+ C Ci ji j<∑ + w(v)2 =12C w v w v C C w viikiikiik+ + − + = = =∑ ∑ ∑( ) ( ) ( )121212 .CONCLUSIONSFrom the above results, a recursive formula for theHosoya-Wiener polynomial of a tree can be deduced.Theorem 5. – Let v1, v2, ..., vk be the sons of a vertex v ina rooted tree T with Hosoya-Wiener polynomials W(Tv1, w,l; x), W(Tv2, w, l; x), ..., W(Tvk, w, l; x) and with partial78 B. ZMAZEK AND J. @EROVNIKCroat. Chem. Acta 80 (1) 75¿80 (2007)Hosoya-Wiener polynomials W x W xv v 1 2( ), ( ), ..., W xvk ( ).Let Tu denote the subtree of T with root u. Then, the Ho-soya-Wiener polynomial of the subtree rooted at v is:W(Tv, w, l; x) = W x W x W w xuu V Tv vvvii( ) ( ) ( , , ; )( )∈∑ ∑= + T lRecall that Wv(x) can be computed from partialpolynomials of the sons of v by Lemma 3. Therefore theformula also gives a recursive algorithm:ALGORITHM A ((Tv, v): rooted tree with root)(i) for all sons vi of v run A(Tvi, vi) (compute Hoso-ya-Wiener polynomial W(Tvi, w, l; x) and partial Hoso-ya-Wiener polynomial W xvi ( ))(ii) combine the results and return W(Tv, w, l; x) (andW xv ( )).Given a tree T with an arbitrary vertex r, runningA(T, r) returns the Hosoya-Wiener polynomial W(T, w,l; x). Another way of writing the algorithm without re-cursion is:ALGORITHM B(i) given a weighted tree T, choose any vertex andcall it the root(ii) order the vertices in the BFS order (breadth firstsearch from the root)(iii) visit the vertices in the reverse BFS order andcompute the W xv ( )(iv) compute (in any order) Wv(x) and sum up overall vertices that are not leaves. The sum equals W(T, w,l; x).The correctness of the two algorithms is a straight-forward consequence of the results proved in the previoussection. We omit the details.Theorem 6. – The algorithms A and B give the Hosoya-Wiener polynomial on a weighted tree T.Remark. If arbitrary edge weights are allowed, then thefunction W(G, w, l; x) is generally not a polynomial.Furthermore, one can find examples where up to n(n –1) / 2 different values for distances appear. Hence, a simpleanalysis yields the complexity estimate O(n3), which isno better than the complexity of computing all distances ina weighted graph with standard algorithms (for example,p. 200 of Ref. 17). It does not seem likely that the pre-sent approach would give a competitive way of comput-ing the Hosoya-Wiener polynomial on trees with arbitra-ry weights.On the other hand, we believe that it is possible togeneralize the approach outlined here to cacti, which maybe an interesting avenue of further research.Example. Given a tree, let us choose one vertex and drawthe tree rooted at the vertex as shown in the Figure.Next, compute the partial Hosoya-Wiener polyno-mials:Finally, compute the vertex contributions and theirsum, giving the result.Acknowledgement. – Supported in part by the Ministry ofHigher Education, Science and Technology of Slovenia.THE HOSOYA-WIENER POLYNOMIAL OF WEIGHTED TREES 79Croat. Chem. Acta 80 (1) 75¿80 (2007)Figure 1.1. Example of weighted tree T.Figure 1.2. Partial Hosoya-Wiener polynomials on vertices of T.4x4 + 4x3 + 10x2 + 2x2x2 + x x2 + 4xW(x) = 2x7 + 9x6 + 7x5 + 9x4 + 11x3 + 17x2 + 7x + 20Figure 1.3. Contribution of each vertex v to the Hosoya-Wienerpolynomial W(x) of T.2x7 + 9x6 + 7x5 + 9x4 + 5x3 + 10x2 + 2x + 42x3 + 2x2 + x + 1 4 x3 + 5x2 + 4x + 1REFERENCES1. H. Wiener, J. Am. Chem. Soc. 69 (1947) 2636–2638.2. D. H. Rouvray, Should we have designs on topological indi-ces?, in: B. B. 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U slu~aju stabala s korijenom Hosoya-Wienerov polinom izra`en je kaozbroj doprinosa ~vorova. Dana je tako|er rekurzivna formula za ra~un Hosoya-Wienerovog polinoma ute`enihstabala.80 B. ZMAZEK AND J. @EROVNIKCroat. Chem. Acta 80 (1) 75¿80 (2007)

The Hosoya-Wiener Polynomial of Weighted Trees

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