The Wiener Index of Trees with Prescribed Diameter

Transcription

1 ± Dec., 011 Operations Research Transactions Vol.15 No.4 The Wiener Inex of Trees with Prescribe Diameter XING Baohua 1 CAI Gaixiang 1 Abstract The Wiener inex W(G) of a graph G is efine as the sum of G(u, v) over all pairs of vertices, where G(u, v) is the istance between vertices u an v in G. In this paper, we characterize the tree with thir-minimum Wiener inex an introuce the metho of obtaining the orer of the Wiener inices among all the trees with given orer an iameter, respectively. Keywors Wiener inex, iameter, tree, istance Chinese Library Classification O3 010 Mathematics Subject Classification 90C70, 90C05 ÊÈ ÎÇ Ç Wiener 1 1 G «Wiener û ľ u, v «G(u, v). Æ Á «¾ ¹Ä µ Wiener û«¹«Å ¹«Wiener û «ËÍÆ Wiener º À ÉÏÌ O1 ÉÏÌ 90C70, 90C05 0 Introuction The Wiener inex of a graph G, efine in [1], is W(G) = G (u, v), u,v V (G) where G (u, v) is the istance between u an v in G an the sum goes over all the pairs of vertices. Since the Wiener inex was introuce by Wiener in 1947 [1], numerous of its chemical applications were reporte an its mathematical properties were unerstoo (see [1-5]). Recently, fining the graphs with minimum or maximum topological inices incluing Wiener inex attracte the attention of many researchers an lots of results are achieve(see ² * This work was supporte by Natural Science Founation of Department of Eucation of Anhui Province (KJ011Z36). 1. School of Mathematics an Computational Science, Anqing Teachers College, Anqing 46133, China; ³ ¼ ½ ³ Å Corresponing author

2 4 ± The Wiener Inex of Trees with Prescribe Diameter 37 [6-1]). For example, Liu an Pan [6] obtaine the graphs with the minimum an seconminimum Wiener inices among all graphs with given orer an iameter. Deng an Xiao [8] gave the maximum Wiener polarity inex of trees with n vertices an k penants. Chen an Zhang [9] characterize all extremal graphs which attain the lower boun of Wiener inex of unicyclic graphs of orer n with girth an the matching number β 3g/. Lin [11] characterize the graphs with the minimum an maximum Wiener inices among all graphs with given orer an clique number. More results of the Wiener inex, we refer the reaers to []. All graphs consiere in this paper are finite, simple an unirecte. For a vertex x of a graph G, we enote the egree of x by (x). The iameter of a graph G, iam(g), is the maximum istance between any two vertices of G. The istance of a vertex x, D G (x), is the sum of istances between x an all other vertices of G. The orer of a graph G is enote by n(g). Let P n an S n enote the path an star of orer n, Tn 3,1 1 is a tree of orer n obtaine from the star S n 1 by attaching a penant vertex to one penant vertex of S n 1. In orer to formulate our results, we efine some trees as follows. We use G = P v1v v k (T 1, T,, T k ) to enote the tree of orer n obtaine from a path P = v 0 v 1 v by ientifying a tree T i to vertex v i of P (1 i k, 1 k 1), where v i is the root of T i, let l k = n(t k ) 1 an l = l 1 + l + + l k, then l = n 1. If T i = Si, let G 1 = P v1v v k (S l1+1, S l+1,, S lk +1), the root v i is the center of the star S li+1. Write H 1 = {P vi (T i) : 1 i 1} an T n,,i = P vi (S n ), X n,,i = P vi H = {P vivj H 3 = {P vivjvm (T 1 n 3,1 ); (T i, T j ) : 1 i < j 1} an Z n,,i,j = P vivj (S n 1, S ), (S n, S 3 ); A n,,i,j = P vivj (T i, T j, T m ) : 1 i < j < m 1} an B n,,i,j,m = P vivjvm (S, S n, S ). Some of these graphs are epicte in Fig.1. We follow [3] for other graph-theoretical terminologies an notations not efine here. Fig. 1 Let T be a tree of orer n with iameter. If = n 1, then T = P n ; if = n,then T is a tree obtaine from the path P = v 0 v 1 v by attaching a penant vertex to the vertex v i of P, ((v i ) = ); if n 3 n 5, then the research on Wiener inex of T is easy; an if =, then T = S n. Therefore, in the following, we assume that 3 n 6. Let T n, = {G : G = P v1v v k (T 1, T,, T k ) is a tree with orer n an

3 38 XING Baohua, CAI Gaixiang 15 iameter, 3 n 6}. In this paper, the thir-minimum Wiener inex of tree in the set T n, (3 n 6) is characterize. 1 Lemmas Lemma 1.1 [] Let T be a tree of orer n. Then W(T) W(S n ) an the equality hols if an only if T = S n. By Lemma 1.1, we have the result. Let H be a connecte graph an T l be a tree of orer l with V (H) V (T l ) = {v}. Then W(HvT l ) W(HvS l ), an equality hols if an only if HvT l = HvSl, where v is ientifie with the center of the star S l in HvS l. Lemma 1. [11] Let H, X, Y be three connecte, pairwise isjoint graphs. Suppose that u, v are two vertices of H, v is a vertex of X, u is a vertex of Y. Let G be the graph obtaine from H, X, Y by ientifying v with v an u with u, respectively. Let G 1 be the graph obtaine from H, X, Y by ientifying vertices v, v, u, an let G be the graph obtaine from H, X, Y by ientifying vertices u, v, u. Then W(G 1 ) < W(G) or W(G ) < W(G). Lemma 1.3 [6] Let P = v 0 v 1 v be a path of orer + 1. Then D p (v j ) = j j + +, for 1 j 1. Moreover, if 1 i < j /, then D p (v i ) > D p (v j ), an if / i < j 1, then D p (v i ) < D p (v j ). Lemma 1.4 Let G = P v1v v k (T 1, T,, T k ) be a tree of orer n with iameter, then W(G) = ( + 1)3 ( + 1) 6 + k k 1 w i + + k j=i+1 k l i D P (v i ) + k W(T i ) (l i w j + l j w i + l i l j P (v i, v j )), where l i = n(t i ) 1, w i = D Ti (v i ) is the sum of istances between v i an all other vertices of T i. Proof In orer to verify the result, we ivie the vertices of G into four cases: (1) x, y V (P ), the sum of istances between x an y, enote by W 1, then W 1 = ( + 1)3 ( + 1). 6

4 4 ± The Wiener Inex of Trees with Prescribe Diameter 39 () x, y V (T i v i ), 1 i k, the sum of istances between x an y, enote by W, then k W = (W(T i ) w i ), where w i = D Ti (v i ) is the sum of istances between v i an all other vertices of T i. (3) x V (T i v i ), y V (T j v j ), 1 i k, 1 j k an i j, the sum of istances between x an y, enote by W 3,then hence x V (T i v i) y V (T j v j) G (x, y) = Ti (x, v i ) + P (v i, v j ) + Tj (v j, y), G (x, y) = W 3 = = x V (T i v i) y V (T j v j) x V (T i v i) ( Ti (x, v i ) + P (v i, v j ) + Tj (v j, y)) (l j Ti (x, v i ) + w j + l j P (v i, v j )) = l j w i + l i w j + l i l j P (v i, v j ), k 1 k j=i+1 (l i w j + l j w i + l i l j P (v i, v j )). Where l i = n(t i ) 1. (4) x V (T i v i ), y V (P ), 1 i k, the sum of istances between x an y, enote by W 4, then Therefore, W 4 = = = = k y V (P ) x V (T i v i) k y V (P ) x V (T i v i) k y V (P ) G (x, y) (l i G (v i, y) + w i ) k (l i D P (v i ) + ( + 1)w i ), ( G (x, v i ) + G (v i, y)) W(G) = W 1 + W + W 3 + W 4 = ( + 1)3 ( + 1) 6 + k k 1 w i + + j=i+1 k l i D P (v i ) + k W(T i ) k (l i w j + l j w i + l i l j P (v i, v j )),

5 40 XING Baohua, CAI Gaixiang 15 where l i = n(t i ) 1, w i = D Ti (v i ), i = 1,,, k. Lemma 1.5 [6] Let T be a non-trivial connecte graph an let v V (T). Suppose that two paths P = vv 1 v v k, Q = vu 1 u u m of lengths k, m(k m 1) are attache to T by their ens vertices at v, respectively, to form T k,m. Then W(T k,m) < W(T k+1,m 1). Main results In this section, we will give the tree with the thir-minimum Wiener inex in the set T n, (3 n 6). Liu an Pan [6] obtaine the tree with the minimum Wiener inex is T n,, an the tree with the secon-minimum Wiener inex is Z n,,, in the set +1 T n, (3 n 3). In fact, by Lemma 1.1, , we have in the set H 1, the Wiener inex is minimize by the tree T n,, an in the set H, the Wiener inex is minimize by the tree Z n,,, +1. Lemma.1 In the set H with 3 n 6, (1) for is even, the secon-minimum Wiener inex of tree is Z n,, 1, or A n,,, +1 ; () for is o, the secon-minimum Wiener inex of tree is Z n,, +1, +. Proof By lemmas in section 1, we get the tree with the secon-minimum Wiener inex in the set H maybe Z n,, v 1, or A n,,, +1 or Z n,, +1, or P v +1 + (Tn 4,1, 1 S ). By Lemma 1.3 an 1.4, it is easy to obtain that W(Z n,, 1, ) W(Z n,, +1, + ) = (n )( 4 if is even, the result is 4 < 0; if is o, the result is (n 3) > 0. ) + ( 4 4 W(Z n,, 1, ) W(A n,,, +1 ) = (n )( ), if is even, the result is 0; if is o, the result is n > 0. W(Z n,, v 1, ) W(P v +1 (Tn 4,1 1, S )) = (n 1)( ), if is even, the result is < 0. W(Z n,, +1, + ) W(A n,,, +1 ) = (n )( ) + 4, ),

6 4 ± The Wiener Inex of Trees with Prescribe Diameter 41 if is o, the result is (n 6) 0. v W(Z n,, +1, + ) W(P v +1 (Tn 4,1 1, S )) = (n 1)( ) + 4, if is o, the result is 5 n < 0. Thus the result hols. Lemma. Let G T n, \(H 1 H ), 3 n 6. Then W(G) W(B n,, 1,, +1 ), with equality if an only if G = B n,, 1,, +1. Proof Let P = v 0 v 1 v 1 v is a path of length in G with (v 0 ) = (v ) = 1. Let V = {v i : (v i ) 3, 1 i 1}. Since n + 3, V φ. We iscuss in two cases. Case 1 V 3. In this case, we first obtain a tree T 1 = G1, by Lemma 1.1, W(G) W(T 1 ) an equality hols if an only if G = T 1. SinceG / (H 1 H ), by Lemma 1., we can obtain a tree T = Bn,,i,j,m such that W(T 1 ) > W(T ) = W(B n,,i,j,m ). Hence W(G) W(T 1 ) > W(T ) = W(B n,,i,j,m ). Case V = 3 an G = B n,,i,j,m. In this case, by Lemma 1.4, we have W(B n,,i,j,m ) = ( + 1)3 ( + 1) + D P (v i ) + D P (v m ) + (n 3)D P (v j ) 6 + (n 3)(n + 1) + (n 1) P (v i, v m ) + (n 3) P (v i, v j ) + (n 3) P (v m, v j ), by Lemma 1.3, it is easy to proof that when v i = v 1,v j = v, v m = v +1, the Wiener inex of B n,,i,j,m achieve the minimum. By case 1 an case, the proof of the Lemma is complete. It is easy to proof that the secon-minimum Wiener inex among all the trees of orer v n is Tn 3,1 1 an W(Tn 3,1) 1 = (n )(n + 1). Therefore, let H P W(H) W(X n,, ) > W(T n,, ). (T ), then By Lemma 1.3 an 1.4, we have the trees with the secon an thir-minimum Wiener inices in the set H 1 maybe T n,, 1, T n,, or X n,,. Lemma.3 In the set H 1 H {B n,, 1,, }(3 n 6), +1 (1) for is even, the thir-minimum Wiener inex is Z n,, 1, or A n,,, (n < + 1); +1 X n,, (n > + 1);

7 4 XING Baohua, CAI Gaixiang 15 () for is o, the thir-minimum Wiener inex is Z n,, +1, +. Proof Let T H 1 H {B n,, 1,, +1 } is the tree with the thir-minimum Wiener inex. By Lemma.1 an., T maybe T n,, 1 or X n,, or T n,, or Z n,, +1, or Z + n,, 1, or B n,, 1,, +1. By Lemma 1.3 an 1.4, for is even, W(Z n,, 1, ) W(X n,, ) = (n )(1 + ) + 1 = n 1, if n > + 1, the result is> 0; if n < + 1, the result is< 0. W(Z n,, 1, ) W(T n,, ) = (n ) < 0, for is o, W(Z n,, 1, ) W(B n,, 1,, ) = < 0, +1 W(Z n,, +1, + ) W(B n,, 1,, ) = (n ) < 0, +1 W(Z n,, +1, + ) W(X n,, ) = 3 < 0, W(Z n,, +1, + ) W(T n,, ) = (n ) < 0. 1 Thus the lemma is prove. Theorem.4 Let (3 n 6). Then (1) for is even, or () for is o, G T n, \{T n,,, T n,, 1 Z n,,, +1 }, W(G) W(Z n,, 1, ) W(G) W(A n,,, ) (n < + 1), +1 W(G) W(X n,, ) (n > + 1); W(G) W(Z n,, +1, + ). Proof Let P = v 0 v 1 v 1 v is a path of length in G with (v 0 ) = (v ) = 1. Let V = {v i : (v i ) 3, 1 i 1}. Since n + 3, V φ. We consier three cases. Case 1 V 3.

8 4 ± The Wiener Inex of Trees with Prescribe Diameter 43 In this case, by Lemma. an.3, we have an for is even, for is o, So, for is even, an for is o, Case V =. W(G) W(B n,, 1,, +1 ), W(B n,, 1,, +1 ) > W(Z n,, 1, ); W(B n,, 1,, +1 ) > W(Z n,, +1, + ). W(G) > W(Z n,, 1, ) W(G) > W(Z n,, +1, + ). In this case, by Lemma.1, we have for is even, for is o, Case 3 V = 1. W(G) W(Z n,, 1, ) or W(G) W(A n,,, +1 ); W(G) W(Z n,, +1, + ). In this case, by Lemma.3, we have for is even, when n > + 1, an when n < + 1, For is o, W(G) W(X n,, ), W(G) W(Z n,, 1, ). W(G) W(Z n,, +1, + ). Accoring to the above metho, if we want to obtain the orer of trees in the set T n, (3 n 6) on Wiener inex, we shoul ivie T n, (3 n 6) into some smaller sets H k (1 k 1). An then, by Lemma 1.1, 1.3 an 1.4, we characterize the orer of trees in the set H k (1 k 1) respectively, at last we will obtain the orer of trees in the set T n, (3 n 6) on wiener inex. References [1] Wiener H. Structural etermination of paraffin boiling point [J]. Amer Chem Soc, 1947, 69: 17-0.

9 44 XING Baohua, CAI Gaixiang 15 [] Dobrymin A A, Entringer R, Gutman I. Wiener inex of trees: theory an application [J]. Acta Appl Math, 001, 66: [3] Bony J A, Murty U S R. Graph Theory with Applications [M]. Lonon: Macmillan Press, [4] Dobrymin A A, Gutman I, Klavzar S, et al. Wiener inex of hexagonal systems [J]. Acta Appl Math, 00, 7: [5] Gutman I, Potgieter J H. Wiener inex an intermolecular forces [J]. Serb Chem Soc, 1997, 6: [6] Liu H Q, Pan X F. On the Wiener inex of trees with fixe iameter [J]. MATCH Commun Math Comput Chem, 008, 60(1): [7] Zhang H, Xu S, Yang Y. Wiener inex of toroial polyhexes [J]. MATCH Commun Math Comput Chem, 006, 56: [8] Deng H Y, Xiao H. The maximum Wiener polarity inex of trees with k penants [J]. Applie Mathematics Letters, 010, 3: [9] Chen Y H, Zhang X D. The Wiener Inex of Unicyclic Graphs with Girth an the Matching Number [J]. Mathematics, 011, (): [10] Liu H Q, Lu M. A unifie approach to extremal cacti for ifferent inices [J]. MATCH Commun Math Comput Chem, 007, 58(1): [11] Lin X X. On the extremal Wiener inex of some graphs [J]. OR Transactions, 010, 14(): [1] Deng H Y. The trees on n 9 vertices with the first to seventeenth greatest Wiener inices are chemical trees [J]. MATCH Commun Math Comput Chem, 007, 57():