The Wiener Index for Weighted Trees

Transcription

1 WSEAS TRANSACTIONS o MATHEMATICS Yajig Wag, Yumei Hu The Wieer Idex for Weighted Trees Yajig Wag Departmet of mathematics Tiaji Uiversity Tiaji P. R. Chia wyjwagya@16.com Yumei Hu* Departmet of mathematics Tiaji Uiversity Tiaji P. R. Chia huyumei@tju.edu.c Abstract: The Wieer idex of a graph is the sum of the distaces betwee all pairs of vertices. I fact, may mathematicias have study the property of the sum of the distaces for may years. The later, we foud that these problems have a pivotal positio i studyig physical properties ad chemical properties of chemical molecules ad may other fields. Fruitful results have bee achieved o the Wieer idex i recet years. Most of the research focus o the extreme values ad the correspodig graphs for the o-weighted simple graphs. I this paper, we cosider the edge-weighted graphs. Firstly, we give the exact defiitio of the distaces i edge-weighted graphs. Secodly, we get a useful variat formula of the Wieer idex. The, we take our attetio o edge-weighted trees of order. We get the miimum, the secod miimum, the third miimum, the maximum, the secod maximum values of the Wieer idex, ad characterize the correspodig extremal trees. Key Words: Weighted graph; Tree; Wieer idex; Miimum value; Maximum value; Boud 1 Itroductio 1.1 Backgroud I chemical graph theory, we use vertices to represet atoms, ad use edges coectig two vertices to represet the covalece. The the structure of each molecule ca be expressed as a graph. The molecular topology is defied as the atoms ad the coectio betwee atoms i chemicals, does ot iclude details such as bod agles. Because of some importat parameters i a molecular, such as eergy, key level ad chage desity, are topology relevat essetially. It does make sese that usig a graph to represet a molecular. Ad early half a cetury, the developmet of quatum chemistry widely is largely due to the results that the cocept of graph beig extesive applied. Oe of the major topics i this field is molecular topological idex. Molecular topological idex is already ca express the structure of molecules quatitatively, as a ivariat of the graph ca be used to related the relatioship betwee the molecular structure ad performace. Say simply, a topological idex is a umerical quatity related to a graph that is ivariat uder graph isomorphism, which ca reflect the physical ad chemical properties ad pharmacological properties of molecules, such as boilig poit, water-solubility, the volume ad surface area of molecular, eergy levels, the electio distributio, etc. Topological idex of molecular graphs is oe of the most widely used descriptors i quatitative structure activity relatioships. Quatitative structure activity relatioships (QSAR) is a popular computatioal biology paradigm i moder drug desig, see [], [7] ad [18]. Oe of the most widely kow topological descriptor is the Wieer idex. The Wieer idex, which amed after chemist Wieer [], is defied as the sum of the distaces betwee all pairs of vertices of a coected graph. Namely, the Wieer idex of a graph G is W = W (G) = d G (u, v) {u,v} V (G) We ca omit the subscript if there are o other graphs. The Wieer idex is a well-kow distace-based topological idex itroduced as a structural descriptor for acyclic orgaic molecules. I fact, the property of the sum of the distaces i a graph is oe of the favorite problems i mathematics. May mathematicias have doe a lot of research o it. The we foud that this problem ot oly ispired the iterest of mathematicias, but also has may applicatio. No matter i chemistry, physics ad molecular biology, the Wieer idex has bee highly effective applied, see [11]. The Wieer idex of graphs has bee extesively studied over the past years. I earlier years, the mai task is to study how to calculate the value of the Wieer idex for a certai graph [4],[9]. The with the E-ISSN: Issue 1, Volume 11, December 01 1

2 WSEAS TRANSACTIONS o MATHEMATICS Yajig Wag, Yumei Hu developmet of drug desig, we eed to kow the extremal values ad the correspodig extremal graphs i special kids of graphs. The earliest result is foud i Mathematicias Etriger, Jackso ad Syder [5] firstly showed that the star ad the path have the miimum ad maximum Wieer idex, respectively. At that time, they use the cocept of distace i graphs, which is actually the Wieer idex. May years later, some chemists got the same results. For o-weighted simple graphs, the property of the Wieer idex have bee achieved fruitful results. I 1997, Gutma et al.[10], have made a survey of them. Recetly, Mathematicias have made a lot of experimets ad cojectures o calculatig various kids of graphs, see some examples i [4], [1] [0]. More importatly, there are may ew results o the extremal values of the Wieer idex for some special kids of graphs, especially of trees. For trees with bouded degrees of vertices, Jele ad Triesch [15] foud a family of trees such that W (T ) is miimized. Fischerma et al. [6] characterized the trees which miimize the Wieer idex amog all trees of give order ad maximum degree, ad the trees which maximize the Wieer idex amog all trees of give order. Moreover, the trees miimizig W P (T ) amog all trees T of order ad k leaves are characterized i [17], where k o less tha ad o more tha 1. Hua Wag [1] characterized trees that achieve the maximum ad miimum Wieer idex, give the umber of vertices ad the degree sequece, several algorithms are preseted ad implemeted. At the same time, mathematicias have foud a variety of sesible variatios of the Wieer idex, ad got effective coclusios. The tree that miimizes the Wieer idex amog trees of give maximal degree is studied i [16], the first two largest ad first two smallest modified Wieer idices i are also idetified, respectively. Bolia Liu et al. give the miimum (resp. maximum) Wieer polarity idex of trees with vertices ad maximum degree are give, ad the correspodig extremal trees are determied, where maximum degree o less tha ad o more tha 1. Jua Rada [19] foud the variatio of the Wieer idex uder certai tree trasformatios, which ca be described i terms of coalescece of trees. As a cosequece, coditios for o-isomorphic trees havig equal Wieer idex are preseted. However, The agle of view of former studies are usually limited withi o-weighted graph. Therefore, they oly cosidered the coectio betwee each atom i the molecule, ad igore the characteristics of the atom ad atomic bod, such as the legth of atomic bod. As well kow, these factors play a decisive role i the physical ad chemical properties of molecules. For istace, the legth of atom bod is strogly correlated to the stability of the molecule, which is a ecessary factor i examiig physical ad chemical properties of molecules. Therefore, it is ecessary to research the molecular structure where the legths of the bods take ito cosideratio. Which more relevat to the chemical actual backgroud. The problem ca be trasformed to cosider the topological idices for edge-weighted graphs, where the weight of the edge represets the legth of the atomic bod. 1. Notatios I this paper, we cosider the extreme values of the Wieer idex for edge-weighted trees of order, deote by T. Udefied otatio ad termiology ca be foud i [1].We show that the miimum, the secod miimum, the third miimum, the maximum ad the secod maximum Wieer idex for T, respectively. Ad the correspodig weighted trees reach the extreme values are discussed. Let G be a coected edge-weighted graph with vertex set V (G) ad edge set E(G). For each edge e of G, let ω G (e) be its weight. For clarity of expositio, we shall refer to the weight of a path p i a weighted graph as its legth, deote by L p, L p = ω p (e). e E(p) Similarly the miimum weight of a (u, v)-path will be called the distace betwee u ad v, deote by d G (u, v). The diameter of a graph G, deoted by D(G), is the largest distace betwee two vertices i G. Sice the Wieer idex is cocered with the distace of vertices, the diameter is importat for us i studyig the idex.the Wieer idex of weighted graph G is defied as W = W (G) = d G (u, v) {u,v} V (G) We ca omit the subscript if there is o other graph. I a tree T T, ay two vertices are coected by a uique path. Cosequetly, the Wieer idex of T ca be rewritte as the sum of the legths of p T, where P is the subgraph of which is a path, i.e., W (T ) = L p. Let P i be the set of paths with i p T edges. Hece, we have a variat. W = W (G) = i=1 p P i L p (1) E-ISSN: Issue 1, Volume 11, December 01

3 WSEAS TRANSACTIONS o MATHEMATICS Yajig Wag, Yumei Hu Let e 1, e,, e 1 be the edge sequece i T, the correspodig edge weights sequece is a 1, a,, a 1. We deote by B the sum of all the edge weights i T. For a graph G, let d(u) be the degree of a vertex u i G, ad N(u) be the eighbor set of u. If d(u) = 1, the u is said to be a pedet vertex i G, ad the edge icidet to u is referred to as pedet edge. The eighbor of pedet vertex is called a support vertex. Deoted by P ad S, the weighted path ad the weighted star o vertex, respectively. Note that the weighted star is uique, but the weighted path is ot uique for differet arragemet of edgeweights. We deote by T i (i = 0, 1,,..., 3 ) the set of trees with i o-pedet edges. Thus T is sorted by the umber of o-pedet edges. Namely, T = 3 i=0 T i. Obviously, there is oly star S i T 0, o matter what weights the edges are give. Clearly T 1 is the set of the trees with two o-pedet vertices v i1 ad v i adjacet to i pedet vertices ad j pedet vertices, respectively, which is deoted by T 1 ij (1 i, j 3, i j = ) i detail (see Figure. 1). a 1 a i a a a i1 i3 v i1 v i a i a 1 Figure 1: T 1 ij The miimal values of the Wieer idex for T Firstly, let s itroduce a useful trasformatio. Suppose T T with at least oe o-pedet edge, say e = uv. Let Q 1 = {w w ad u are i the same compoet of T \uv}, ad Q = {w w ad v are i the same compoet of T \uv}. By cotractig uv to v ad addig a pedet vertex u, with ω T (uv) = ω T (uv), ad ω T (f) = ω T (f) for ay f E(T \uv), we get a ew tree T. This trasformatio is deoted by the f(u, v)-trasformatio o T (see Fig. ). f(u, v) = Q 1 Q T u v v Q 1 Q u Figure : f(u, v)-trasformatio The followig lemma is crucial. Lemma 1 Suppose T T with at least oe opedet edge e = uv. T is the tree get from T by f(u, v)-trasformatio. The we have W (T ) > W (T ). Proof: Let w 1 be a vertex i Q 1 ad w be a vertex i Q. Sice W (T ) = d(w 1, u) d(w, u) w 1 Q 1 w Q d(w 1, v) d(w, v) w 1 Q 1 w Q d(w 1, w ) d(v, u) w 1 Q 1,w Q d(x, y) d(x, y). {x,y} Q 1 {x,y} Q Assume that ω(uv) = a (a > 0). The by straightforward observatio, we have that the distace betwee w 1 ad w decreases by f(u, v)- trasformatio. There are altogether Q 1 Q such pairs. So the total reductio is Q 1 Q a. The distace betwee w 1 ad u icreases a by f(u, v)- trasformatio. The total is Q 1 a. The distace betwee w 1 ad v decreases a. The total is Q 1 a. Therefore the sum of d(w 1, u) d(w 1, v) does ot chage. Similarly, the sum of d(w, u) d(w, v) does ot chage, too. Ad the distaces betwee all pairs of the remaiig vertices do ot chage. Cosequetly, W (T ) W (T ) = Q 1 Q a, By the defiitio of Q 1 ad Q, we have Q 1 1 ad Q 1. Hece we get W (T ) > W (T ). T E-ISSN: Issue 1, Volume 11, December 01 3

4 WSEAS TRANSACTIONS o MATHEMATICS Yajig Wag, Yumei Hu The followig theorem shows the sharp lower boud of W (T ) i weighted trees. Theorem The weighted star S has the miimum value of the Wieer idex i weighted trees of order, ad W (S ) = ( 1)B, where B = 1 a i. i=1 Proof: By cotradictio, we assume that weighted tree T S has the miimum Wieer idex. The T has at least a o-pedet edge. By f(u, v)- trasformatio, we get a ew tree T. From Lemma1, we have W (T ) > W (T ), a cotradictio. We coclude that T = S. Now let s compute the Wieer idex of weighted star S. Sice the diameter of S is. Let p be a path i S, the either p P 1 or p P. Accordig to variat formula Eq.(1), we have W (S ) = L p L p p P 1 p P Obviously, p P 1 L p = 1 i=1 a i = B Now we compute L p. Sice for ay edge p P e E(S ), there are paths coverig e i P. So that ω(e) will accumulate times i computig L p.therefore, L p = ( )B. P P p P From the above, W (S ) = ( 1)B. The followig two theorems cosider the secod miimum value ad the third miimum value of W (T ) for weighted trees. Firstly, Let s itroduce some otatios. Deote by Si,j mi, the weighted tree belogs to Tij 1 ad the weight of the uique opedet edge is a mi = mi {a i}. Let a sec be the 1 i 1 secod miimum umber of a 1, a,, a 1. Similarly, deote by Si,j sec(sthird i,j ), the tree with the uique o-pedet edge weights the secod (third) miimum umber of a 1, a,, a 1. Theorem 3 The weighted tree S1, 3 mi has the secod miimum value of the Wieer idex for weighted trees of order, ad W (S1, 3 mi ) = ( 1)B( 3)a mi, where a mi = mi {a i} 1 i 1 Proof: Suppose T T has the secod miimum value of Wieer idex. By Theorem, we have T S. Thus, T has at least oe o-pedet edge. If T has at least two o-pedet edges. By f(u, v)- trasformatio o T, we get T, ad T S. By Lemma 1, we have W (T ) > W (T ), cotradictig to the choice of T. So there is oly oe o-pedet edge i T, i.e., T T 1 = Tij 1, (1 i, j 3,i j = ). We shall compute the Wieer idex of Tij 1. For the sake of coveiece, we regard T 1 ij as two stars joiig together (see Figure. 3). a 1 a i a a i3 a i1 a i1 a i a 1 Figure 3: Accordig to formula of W (S ), for ay T T 1 ij, = W (T ) = 3 L p i=1 p P i L p L p L p p P 1 p P p P 3 = B i (a 1 a a i1 ) ( i ) (a i1 a i a 1 ) j (a 1 a a i ) i j a i1 i (a i a i3 a 1 ) = ( 1)B i j a i1 () Sice ( 1)B is a costat, the W (T ) is decided by the value of i j a i1. Now we discuss the value of i j a i1. Sice 1 i, j 3, ij =. The accordig to the priciple of rearragemet iequality, we may coclude that ( 3) a mi i j a i1 with equality if ad oly if i = 1, j = 3, a i1 = a mi = mi {a i}, amely T = S1, 3 mi. The proof 1 i 1 is completed. Theorem 4 If a sec ( 4) 3 a mi, the the the weighted tree S, 4 mi has the third miimum value of the Wieer idex for weighted trees of order ; ad if a sec ( 4) 3 a mi, the the weighted tree S1, 3 sec has the third miimum value of the Wieer idex for weighted trees of order. Proof: Suppose T T has the third miimum value of Wieer idex. By Theorem, we have T S. E-ISSN: Issue 1, Volume 11, December 01 4

5 WSEAS TRANSACTIONS o MATHEMATICS Yajig Wag, Yumei Hu Thus, T has at least oe o-pedet edge. Let m be the umber of the o-pedet edges i T. If m 3, by f(u, v)-trasformatio o T, we get a ew tree T, T S ad T S1, 3 mi. From Lemma1, we have W (T ) > W (T ), cotradictig to the choice of T. If m =, i.e., T T. The T (see Figure.4 ) has three o-pedet vertices, says u,v ad w, adjacet to r, s ad t pedet vertices, where s 0, r, t 1, r s t = 3. a 1 a sr3 a r a sr1 a a a sr4 r1 a sr u v w a r a 1 Figure 4: T T Now let s compute the Wieer idex of weighted tree T. 4 W (T ) = L p i=1 p P i = L p L p L p L p p P 1 p P p P 3 p P 4 It is obvious that L p = B p P 1 Ad we have L p p P = r (a 1 a a r1 ) (s 1) (a r1 a rs ) t (a rs a 1 ) p P 3 L p = (s 1) (a 1 a a r ) a r1 r(s 1) (s 1) (a rs3 a 1 t (a r1 a rs1 ) (s 1) t a rs p P 4 L p = t (a 1 a a r ) r (a rs3 a 1 ) r t (a r1 a rs ) By simplificatio, we have W (T ) = ( 1)Br( r)a r1 t( t)a rs. Sice S, 4 mi T 1, accordig to Eq. (), we have ij W (S mi, 4) = ( 1)B ( 4)a mi. As we kow that S mi, 4 S, S mi, 4 Smi 1, 3. W (T ) W (S, 4) mi [r( r) t( t)]a mi ( 4)a mi ( 3)a mi ( 4)a mi > 0 Namely, for ay T T, we have W (T ) > W (S mi, 4). It cotradicts to the choice of T. From above discussio, we have m = 1, i.e.,t T 1. Accordig to Eq. (1), we have W (S mi 1, 3) < W (S mi, 4) < < W (S mi Moreover, it is clear that W (S mi 1, 3) < W (S sec 1, 3) < W (S third 1, 3)., ). We ca make the coclusio that the weighted tree with the third miimum value of the Wieer idex is. Furthermore, sice S sec 1, 3 or Smi, 4 W (S sec 1, 3) w(s mi, 4) = ( 3)a sec ( 4)a mi. The if a sec > ( 4) 3 a mi, S, 4 mi has the third miimum value of the Wieer idex; if a sec < ( 4) 3 a mi, S1, 3 sec has the third miimum value of the Wieer idex. 3 The maximal values of the Wieer idex for T Let P 1 = e 1 e e 1 be the classes of the weighted paths with edge weight a 1, a,, a 1. respectively. We deote by P the weighted path, which the distributio of the weights is uimodal, amely, ad a 1 a a 1 a a a 1 a a 1. Firstly, let s itroduce a useful trasformatio. Suppose T T. Let v 0 v 1 v v d be the logest path E-ISSN: Issue 1, Volume 11, December 01 5

6 WSEAS TRANSACTIONS o MATHEMATICS Yajig Wag, Yumei Hu i T, ad v i be the first vertex with degree more tha. Suppose N(v i ) \ {v i 1, v i1 } = {u 1, u,, u j }. By deletig the edge v i u k ad addig the edge v 0 u k (k = 1,,, j), we get a ew tree T. The let ω T (v i u k ) = ω T (v 0 u k ) for k = 1,,, j, ad ω T (f) = ω T (f), for ay f E(T \v i u k ). This trasformatio is deoted by the g-trasformatio o T (see Fig.5). Q i Q i1 T v 0 v 1 v v i 1 v i v i1 Q i Q i1 T v 0 v 1 v v i 1 v i v i1 Figure 5: g-trasformatio Lemma 5 Let T be a weighted tree of order. If T P 1 by g-trasformatio o T, we get T. The W (T ) < W (T ). Proof: Let v 0 v 1 v v d be the logest path i T T, ad v i be the first vertex with degree more tha, sice T P 1. Let Q 1 = {w w ad v i are i the same compoet of T \{v i 1 v i, v i v i1 }}, ad Q = {w w ad v i1 are i the same compoet of T \{v i v i1 }}. Sice W (T ) = i i1 k=0 j=k1 d(v k, v j ) i1 j=0 w 1 Q i d(w 1, v j ) i1 d(w, v j ) d(x, y) j=0 w Q i1 {x,y} Q i d(w 1, w ) d(x, y). w 1 Q i,w Q i1 {x,y} Q i1 We cosider the chage of the distaces betwee all pairs of vertices after g-trasformatio o T. If w 1 Q 1, w Q, the the value of d(w 1, w ) icreases. Note that the value of d(w 1, v j ) does i ot j=0 chage, ad the distaces betwee remaiig pairs of vertices do ot chage. Cosequetly, W (T ) < W (T ). The proof is completed. The followig theorem shows the maximum value of W (T ) for weighted trees. Theorem 6 The weighted path P has the maximum value of the Wieer idex for weighted trees of order. Moreover, W (P) = 1 k( k)a k. k=1 Proof: By cotradictio, we assume that T T has the maximum Wieer idex. If T P 1, by g- trasformatio o T, we get a ew tree T. By Lemma 5, we have W (T ) < W (T ), a cotradictio. So T P 1 (see Figure.6). a 1 a a k a a 1 e 1 e e k e e 1 Figure 6: T P 1 We ow compute the Wieer idex of T. Let p be a path coverig the edge e k (k = 1,,, 1). There are k differet choices o the left of e k i p, while k o the right. So there are k( k) paths coverig the edge e k. So the accumulatio times of the weight of e k is k( k). whe calculatig the Wieer idex of T. Therefore, W (T ) = 1 k=1 k( k)a k. Now we cosider the maximum value of W (T ). Let (k) = k( k), k = 1,,, 1, we have (1) () ( ), ( ) ( 1 ) ( 1). Accordig to the priciple of rearragemet iequality, we may coclude that the value of W (T ) reaches its maximum whe the edge weights sequece satisfies ad a 1 a a 1 a a a 1 a a 1. i.e., T = P. Ad W (P) = 1 k=1 k( k)a k. Before cosiderig the secod maximum value of the wieer idex, we itroduce a special kid of weighted trees. We deote by T, the set of the E-ISSN: Issue 1, Volume 11, December 01 6

7 WSEAS TRANSACTIONS o MATHEMATICS Yajig Wag, Yumei Hu weighted trees of order, with oly oe 3-degree vertex (which also called brachig vertex), three 1- degree vertices ad the other vertices are of degree. The diameter of T, deote by D(T ), is the maximum distace i T. Let T 1 = {T : T T, D(T ) = }, T = {T : T T, D(T ) < }. The T = T 1 T. Suppose the weighted tree H T, let P = v 0 v 1 v,, v D(H), the logest path i H, ad d H (v i ) = 3. Let Q = {u 1, u,, u j } = {w V (H) w ad v i are i the same compoet of H\{v i 1 v i, v i v i1 }}, where j mi{i, D(H) i}. By deletig the edge u 1 u ad addig the edge v 0 u 1, we get a ew tree H. Let ω H (u 1 u ) = ω H (v 0 u 1 ), ad ω H (f) = ω H (f) for ay f E(H\u 1 u ). The trasformatio from H to H is deoted the h-trasformatio o H (see Fig.7). u 1 u u 3 u j H v 0 v 1 v v i 1 v i v i1 v D(H) 1 v D(H) u u 3 u j H u 1 v 0 v 1 v v i 1 v i v i1 v D(H) 1 v D(H) D(H) k=i1 d(u 1, v k ) j 1 j r=k1 d(u k, u r ) We cosider the chage of the distaces betwee all pairs of vertices after h-trasformatio o H. j First of all, the value of d(u 1, u k ) i d(u 1, v k ) does ot chage. k=0 The distace betwee u 1 ad v k (k = i 1, i,..., D(H) ) icreases, sice p = v 0 v 1 v,, v D(H) is the logest path. The distaces betwee all pairs of the remaiig vertices do ot chage. Cosequetly, we have W (H) < W (H ). The proof is thus completed. Now let s itroduce aother useful trasformatio. Suppose the weighted tree H T 1, let p = v 1 v,, v 1 be the path i H. Let v i be the brachig vertex, i.e., d(v i ) = 3. Without lose of geerality, we ca assume i 1. By deletig edge v iv ad addig the edge v i 1 v, we get a ew tree H. Let ω H (v i v ) = ω H (v i 1 v ), ad ω H (f) = ω H (f) for ay f E(H\v i v ). The trasformatio is deoted by the ψ-trasformatio o (see Fig. 8). v 1 v 1 v 3 v i 1 v i v i1 v 3 v v 1 v v v 1 v v 3 v i 1 v i v i1 v 3 v v 1 H H Figure 7: h-trasformatio Figure 8: ψ-trasformatio Lemma 7 Suppose H T. By h-trasformatio o H, we get H. The we have W (H) < W (H ). Proof: Sice W (T ) = D(H) 1 D(H) k=0 r=k1 j d(u 1, u k ) d(v k, v r ) i d(u 1, v k ) k=0 j D(H) r=0 d(u k, v r ) Lemma 8 Suppose H T 1. By ψ-trasformatio o H, we get a ew tree H. The we have W (H) < W (H ). Proof: Let a (a > 0) be the weight of edge v i 1 v i. Sice W (T ) = 1 k=1 r=k1 i 1 d(v k, v r ) d(v j, v ) j=1 E-ISSN: Issue 1, Volume 11, December 01 7

8 WSEAS TRANSACTIONS o MATHEMATICS Yajig Wag, Yumei Hu 1 j=i d(v j, v ) Now we cosider the chage of the distaces betwee all pairs of vertices. The distace betwee the vertex v j ad v decreases a, for j = 1,,..., i 1. There are altogether i 1 such pairs. So the total decrease is (i 1)a. The distace betwee the vertex v j ad v icreases a, for j = i, i 1,..., 1. There are altogether i such pairs. So the total icrease is ( i)a. The distaces betwee all pairs of the remaiig vertices do ot chage. Cosequetly, W (H) W (H ) = (i 1)a ( i)a = (i 1)a Sice i 1 ad a > 0, the we have W (H) < W (H ). Deote by H the weighted tree i T 1 with two pedet vertices adjacet to the uique brachig vertex. Let p = v 1 v,, v 1 be the path with D(H ) edges i H with the correspodig weights are a 1, a 3,, a 1, ad the edge v v weights a (see Fig. 9). The weights a 1, a,, a 1 meet the followig coditios: ad a 1 a a 1 a, a a 1 a a 1. v a a 1 v 1 a 3 v a 4 v 3 a a 1 v 4 v 3 v v 1 Figure 9: H Theorem 9 The weighted tree H has the maximum value of the Wieer idex i T. Proof: Suppose H T have the maximum Wieer idex i T. The either H T 1 or H T. If H T. By h-trasformatio o H, we get a ew tree H. By Lemma7, we have W (H) < W (H ). It cotrary to the hypothesis. So H T 1. Let p = v 1 v,, v 1 be the logest path i H, ad d(v i ) = 3. Without lose of geerality, we ca assume i 1. From Lemma 8, we ca coclude that the brachig vertex i H is v. We shall calculate the Wieer idex of H. For the sake of coveiece, we look H as two paths joiig together (see Fig. 10). a 1 a 3 a 4 a a 1 v 1 v v 3 v 4 v 3 v v 1 v a a3 a 4 a a 1 v v 3 v 4 v 3 v v 1 Figure 10: Accordig to Wieer idex formula of P, W (H) = ( ) 3 ( ) 3 ( ) 4 ( a 1 1 a k ) [( 1) k 1](k 1)a k1 1 a k [( 1) k 1](k 1)a k1 1 = ( 1)B 3 a k a 1 a [( 1) k 1](k 1)a k 3 ( k )(k 1)a k1 ( k )(k 1)a k1 = ( 1)B ( 4)a 3 3 ( k )k a k1 = ( 1)B ( 4)a 3 k=4 ( k 1)(k 1)a k E-ISSN: Issue 1, Volume 11, December 01 8

9 WSEAS TRANSACTIONS o MATHEMATICS Yajig Wag, Yumei Hu = ( 1)B ( k 1)(k 1)a k Accordig to the priciple of rearragemet iequality, we may coclude that W (H) will reach its maximum value while the edge weights sequece satisfies a 1 a a 1 a ad a a 1 a a 1. i.e., H = H. Moreover, W (H ) = ( 1)B ( k 1)(k 1)a k The proof is completed. The followig theorem shows the secod maximum value of W (T ) for weighted trees. Theorem 10 Suppose T T has the secod maximum value of the wieer idex, the T P 1 or T = H. Proof: Suppose T T, T P have the secod maximum Wieer idex. If T P 1 ad T T, by g-trasformatio, we ca get weighted tree H. By Lemma5, we have W (H) < W (H ), a cotradictio. So we have either T P 1 or T T. By Theorem9, the weighted tree H has maximum Wieer idex i T.Therefore, if T T has the secod maximum value of the Wieer idex. The T P 1 or T = H. 4 Coclusio We explore a ew class of trees for computig Wieer idices that have ot bee studied before to the best of our kowledge.we itroduced tree operatios ad derived formulas for the Wieer idices uder these operatios. Our mai results are listed as follows. 1. The weighted star S has the miimum value of the Wieer idex i weighted trees of order, ad W (S ) = ( 1)B, where B = 1 a i ; i=1. The weighted tree S1, 3 mi has the secod miimum value of the Wieer idex for weighted trees of order, ad W (S1, 3 mi ) = ( 1)B ( 3)a mi, where a mi = mi {a i} 1 i 1 3. If a sec > ( 4) 3 a mi, the the the weighted tree S, 4 mi has the third miimum value of the Wieer idex for weighted trees of order ; ad if a sec < ( 4) 3 a mi, the the weighted tree S1, 3 sec has the third miimum value of the Wieer idex for weighted trees of order. 4. The weighted path P has the maximum value of the Wieer idex for weighted trees of order. Moreover, W (P) = 1 k( k)a k. k=1 5. Suppose T T has the secod maximum value of the wieer idex, the T P 1 or T = H. Ackowledgemets: The correspodig author, Yumei Hu, is supported by NSFC No Refereces: [1] A. Body, U. S. R. Murty,Graph Theory with Applicatios, New York: Macmilla Press, [] H. Corwi, A. Kurup, R. Garg, H. Gao, Chembioifomatics ad QSAR: a review of QSAR lackig positive hydrophobic terms, Chem. Rev., vol.101, 001, pp [3] A. A. Dobryi, R. Etriger, I. Gutma, Wieer idex of trees theory ad applicatio, Acta. Appl. Math., vol.66, 001, pp [4] A. A. Dobryi, I. Gutma, S. Klavzar, et.al., Wieer idex of hexagoal systems, Acta Appl. Math., vol.7, 00, pp [5] R. C. Etriger, D. E. Jackso, D. A. Syder, Distace i graphs, Czechoslavak Math. J. vol.6, 1976, pp [6] M. Fischerma, A. Hoffma, L. S. Dieter Rautebach, L. Volkma,Wieeridex versus maximum degree i trees, Discrete Appl. Math., vol.1, 00, No.1-3, pp [7] R. Gozalbes, J. Doucet, F. Deroui,Applicatio of topological descriptors i QSAR ad drug desig: history ad ew treds, Curret Drug Targets: Ifectious Disorders, vol., 00, pp [8] I. Gutma, A property of the Wieer umber ad its modificatios. Idia J. Chem., 6A, 1997, [9] I. Gutma, A ew method for the calculatio of the Wieer umber of acyclic molecules, J. Molecular Struct., vol.85, 1993, pp [10] I. Gutma, S. Klavzar, B. Mohar, Fifty years of the Wieer idex. MATCH Commu. Math. Comput. Chem., vol.35, 1997, pp E-ISSN: Issue 1, Volume 11, December 01 9

10 WSEAS TRANSACTIONS o MATHEMATICS Yajig Wag, Yumei Hu [11] I. Gutma, W. Liert, I. Lukovits, ad A. A. Dobryi, Trees with extremal hyper-wieer idex: Mathematial basis ad chemical applicatios, Chem. If. Comput. Sci., vol.37, 1997, [1] I. Gutma, J. Rada, O. Araujo. The Wieer idex of starlike trees ad a related partial order. MATCH Commu. Math. Comput. Chem., vol.4, 000, pp [13] I. Gutma, Y. N. Yeh, S. L. Lee, ad Y. L. Luo,Some recet results i the theory of the Wieer umber, Idia J. Chem., 3A, 1993, pp [14] O. Ivaciuc, QSAR comparative study of Wieer descriptor for weighted molecular graphs, J. Chem. If. Comput. Sci., vol. 40, 000, pp [15] F. Jele, E. Triesch,Superdomiace order ad distace of trees with bouded maximum degree, Discrete Appl. Math., vol.15, 003, No.- 3, pp [16] M. Liu, B. Liu, O the variable Wieer idices of trees with give maximum degree,mathematical ad Computer Modellig, vol.5, 010, No.9-10, pp [17] B. Liu, H. Hou, Y. Huag, O the Wieer polarity idex of trees with maximum degree or give umber of leaves, Computers ad Mathematics with Applicatios, vol.60, 010, No.7, pp [18] Y. C. Marti, 3D QSAR. Curret state, scope, ad limitatios, Perspect. Drug Discovery Des., vol.1, 1998, pp [19] J. Rada, Variatio of the Wieer idex uder tree trasformatios, Discrete Applied Mathematics, vol.148, 005, No., pp [0] S. Wager, O the Wieer idex of radom trees, Discrete Mathematics, vol.31, 01, No.9, pp [1] H. Wag, The extremal values of the Wieer idex of a tree with give degree sequece, Discrete Applied Mathematics, vol.156, 008, No.14, pp [] H. Wieer, Structral determiatia of paraffi boilig poits, J. Am. Chem. Soc., vol.69, 1947, pp E-ISSN: Issue 1, Volume 11, December 01