Research Article The Hyper-Wiener Index of Trees of Order n with Diameter d

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1 Discrete Dynamics in Nature and Society Volume 016, Article ID , 5 pages Research Article The Hyper-Wiener Index of Trees of Order n with Diameter d Gaixiang Cai, 1 Guidong Yu, 1 Jinde Cao,,3 Ahmad Alsaedi, 4 and Fuad Alsaadi 5 1 School of Mathematics & Computation Sciences, Anqing Normal University, Anqing 46011, China Department of Mathematics, Southeast University, Nanjing, Jiangsu 10096, China 3 Department of Mathematics, King Abdulaziz University, Jeddah 1589, Saudi Arabia 4 Faculty of Science, King Abdulaziz University, Jeddah 1589, Saudi Arabia 5 Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 1589, Saudi Arabia Correspondence should be addressed to Jinde Cao; jdcao@seu.edu.cn Received 3 March 016; Accepted 31 August 016 AcademicEditor:JuanR.Torregrosa Copyright 016 Gaixiang Cai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The hyper-wiener index is a kind of extension of the Wiener index, used for predicting physicochemical properties of organic compounds. The hyper-wiener index WW(G) is defined as WW(G) = (1/) u,v V(G) (d G (u, V) +d G (u, V)) with the summation going over all pairs of vertices in G, andd G (u, V) denotes the distance of the two vertices u and V in the graph G. In this paper, we obtain the second-minimum hyper-wiener indices among all the trees with n vertices and diameter d and characterize the corresponding extremal graphs. 1. Introduction Let G be a simple graph of order n with vertex set V(G) and edge set E(G). The distance between two vertices u, V of G, denoted by d G (u, V) or d(u, V), is defined as the minimum length of the paths between u and V in G. Thediameterof agraphg is the maximum distance between any two vertices of G. The Wiener index of a graph G, denoted by W(G), is one of the oldest topological indexes, which was first introduced by Wiener [1] in It is defined as W(G) = u,v V(G) d G (u, V) where the summation goes over all pairs of vertices of G. The hyper-wiener index of acyclic graphs was introduced by Randić in 1993 []. Then Klein et al. [3] extended the definition for all connected graphs, as a generalization of the Wiener index. Similar to the symbol W(G) for the Wiener index, the hyper-wiener index is traditionally denoted by WW(G). The hyper-wiener index of agraphg is defined as WW (G) = 1 ( u,v V(G) d G (u, V) + d G u,v V(G) (u, V)). (1) Let S(G) = u,v V(G) d G (u, V). Then WW (G) = 1 W (G) + 1 S (G). () We denote D G (u) = V V(G) d G (u, V), DD G (u) = V V(G) d G (u, V),andthen W (G) = 1 S (G) = 1 u V(G) u V(G) D G (u), DD G (u). Recently, the properties and uses of the hyper-wiener index have received a lot of attention. Feng and Ilić [4] studied hyper-wiener indices of graphs with given matching number. Feng et al. [5] researched the hyper-wiener index of unicyclic graphs. Feng et al. [6] discussed the hyper- Wiener index of bicyclic graphs. Gutman [7] obtained the relation between hyper-wiener and Wiener index. Xu and Trinajstić [8] discussed hyper-wiener index of graphs with cut edges. M.-H. Liu and B. L. Liu [9] determined trees with (3)

2 Discrete Dynamics in Nature and Society 0 1 i Z n,d,i,j (a) d+1 j d 1 d d i d 1 d T n,d,i (b) d+3 d+ d i d 1 d Y n,d,i (c) d+ d+3 d i d 1 d X n,d,i (d) Figure 1 the seven smallest and fifteen greatest hyper-wiener indices. Yu et al. [10] studied the hyper-wiener index of trees with given parameters. All graphs considered in this paper are finite and simple. Let G be a simple graph with vertex set V(G) = {V 1, V,...,V n } and edge set E(G). ForavertexV V(G), the degree and the neighborhood of V are denoted by d G (V) and N G (V) (or written as d(v) and N(V) for short). A vertex V of degree 1 is called pendant vertex. An edge e=uvincident with the pendant vertex V is called a pendant edge. For a subset U of V(G), letg Ube the subgraph of G obtained from G by deleting the vertices of U and the edges incident with them. Similarly, for a subset E of E(G), wedenotebyg E the subgraph of G obtained from G by deleting the edges of E.If U={V} and E ={uv},thesubgraphsg Uand G E will be written as G V and G uv forshort,respectively.forany two nonadjacent vertices u and V in graph G, weuseg+uv to denote the graph obtained from G by adding a new edge uv. DenotebyS n and P n the star and the path on n vertices, respectively. Atreeisaconnectedacyclicgraph.LetT be a tree of order n with diameter d. Ifd=n 1,thenT P n ;andif d=,thent S n. Therefore, in the following, we assume that 3 d n. Let T n,d = {T : T is a tree with order n and diameter d, 3 d n }. Liu and Pan [11] characterized the minimum and second-minimum Wiener indices of trees in the set T n,d (3 d n ),andyuet al. [10] characterized the minimum hyper-wiener index of all trees on n vertices with diameter d. Motivatedbythese articles, we will characterize the second-minimum hyper- Wiener indices of trees in the set T n,d (3 d n )in this paper.. Preliminaries Lemma 1 (see [8]). Let H, X,andY be three connected graphs disjoint in pair. Suppose that u, V are two vertices of H, V 1 is avertexofx, andu 1 is a vertex of Y. LetG be the graph obtained from H, X,andY by identifying V with V 1 and u with u 1,respectively.LetG 1 be the graph obtained from H, X, and Y by identifying three vertices V, V 1,andu 1,andletG be the graph obtained from H, X, andy by identifying three vertices u, V 1,andu 1.Thenonehas WW (G 1 )<WW(G) (4) or WW (G )<WW(G). ByLemma1,wehavethefollowingresult. Corollary. Let G be a graph and u, V V(G). Suppose that G p,q is the graph obtained from G by attaching p, q pendant vertices to u, V,respectively.Then WW (G p i,q+i ) < WW (G p,q ), 1 i p, (5) or WW (G p+i,q i )<WW(G p,q ), 1 i q. (6) Let G 1,G be two connected graphs with V(G 1 ) V(G )= {V}.LetG 1 VG be a graph with V(G 1 ) V(G ) as its vertex set and E(G 1 ) E(G ) asitsedgeset.wehavethefollowingresult. Lemma 3 (see [8]). Let H be a connected graph, T m be a tree of order m,andv(h) V(T m )={V}. Then WW (HVT m ) WW(HVS m ), (7) and equality holds if and only if HVT m HVS m,wherev is the center of star S m. Lemma 4 (see [6]). Let G be a connected graph of order n, V be a pendant vertex of G,andVw E(G).Then (1) W(G) = W(G V)+D G V (w) + n 1; () S(G) = S(G V)+DD G V (w) + D G V (w) + n 1. By Lemma 4 and the definition of hyper-wiener index, we have the following result. Corollary 5. Let G be a connected graph of order n, V be a pendant vertex of G,and Vw E(G).Then WW (G) =WW(G V) + 1 DD G V (w) + 3 D G V (w) +n 1. (8)

3 Discrete Dynamics in Nature and Society 3 Lemma 6 (see [9]). Let G be a connected graph on n vertices and V be a vertex of G. LetG k,l be the graph obtained from G by attaching two new paths P:VV 1 V V k and Q: Vu 1 u u l of length k and l at V,respectively,whereu 1,...,u l and V 1,...,V k are distinct new vertices. Let G k+1,l 1 =G k,l u l 1 u l + V k u l.ifk l 1,then WW (G k,l ) <WW(G k+1,l 1 ). (9) Let V 1 be a vertex of graph G, andv,...,v t+s, u 0 are distinct new vertices (not in G). Let G be the graph GV 1 P, where P is a new path V 1 V V t+s. Let M t,t+s =G + V t u 0 and M t+i,t+s =G + V t+i u 0,where1 i s. Lemma 7 (see [9]). Suppose G is a connected graph on n vertices or has only one vertex. If t s 1,thenWW(M t,t+s ) WW(M t+i,t+s ),where1 i s. 3. Main Results In this section, we will give the second-minimum hyper- Wiener index in the set T n,d (3 d n ). In order to formulate our results, we need to define some trees as follows. Let T n,d (p 1,...,p d 1 ) be a tree of order n obtained from a path P d+1 = V 0 V 1 V d 1 V d by attaching p i pendant vertices to V i, 1 i d 1, respectively, where n=d+1+ d 1 i=1 p i, p i 0, i= 1,,...,d 1. Denote T n,d,i =T n,d (0,...,0,n d 1,0,...,0)(see Figure 1(b)), and we note that T n,d,i T n,d,d i. Denote Z n,d,i,j =T n,d (0,...,0,n d,0,...,0,1,0,...,0) i 1 j i 1 (see Figure 1(a)). For i d,letx n,d,i be a graph obtained from T n 1,d,i by attaching a pendant vertex to one pendant vertex of T n 1,d,i except for V 0, V d (see Figure 1(d)). Then X n,d,i X n,d,d i.let Y n,d,i be a graph obtained from T d+,d,i by attaching n d pendant vertices to one pendant vertex of T d+,d,i except for V 0, V d (see Figure 1(c)). Then Y n,d,i Y n,d,d i. Yu et al. [10] characterized that T n,d, d/ had the minimum hyper-wiener index of all trees on n vertices with diameter d. In the following, we firstly give possible trees with the second-minimum hyper-wiener indices in the set T n,d (3 d n ). Denote T 1 ={T n,d,i :1 i d 1},T ={X n,d,i : i d },T 3 ={Y n,d,i : i d },andt 4 ={Z n,d,i,j : 1 i<j d 1}. Lemma 8. Let T T n,d.thenww(t) WW(T n,d,i ), or WW(T) WW(X n,d,i ),orww(t) WW(Y n,d,i ),or WW(T) WW(Z n,d,i,j ). Proof. If T T 1, the conclusion is obvious. If T T 1,let P d+1 = V 0 V 1 V d 1 V d be a path of length d in T with d(v 0 )= d(v d )=1.Let V d ={V i :d(v i ) 3,1 i d 1}.Since n d+,v d =0, we consider two cases. Case 1 ( V d ). By Lemma 3, WW (T) WW(T n,d (p 1,...,p d 1 )). (10) i 1 By Corollary, WW(T n,d (p 1,...,p d 1 )) WW(T n,d (0,...,0,q i,0,...,0,q j,0,...,0)) WW(Z n,d,i,j ), i 1 j i 1 where p 1 + +p d 1 =q i +q j. So WW (T) WW(Z n,d,i,j ). (11) Case ( V d = 1). Let V i V d and N(V i )\{V i 1, V i+1 } = {w 1,w,...,w s } with d(w j ),1 j r,andd(w r+1 )= =d(w s )=1.Then r 1as T T 1. Let T i (w j ) be subtrees of T V i which contain w j,and V(T i (w j )) = s j +1, 1 j r. By Lemma 3, we can obtain a tree T 1 obtained from T d+s+1,d,i by attaching s j pendant vertices to w j, 1 j s,respectively, such that WW(T) > WW(T 1 ).ByCorollary,wecanobtain atreet T T 3 such that WW(T 1 ) WW(T ). So WW(T) WW(X n,d,i ),orww(t) WW(Y n,d,i ). Lemma 9. Let P=V 0 V 1 V d be a path of order d+1.then D P (V j )= j dj+d +d, DD P (V j ) (1) = 6j +6dj 6d j 6dj + d 3 +3d +d, 6 for 1 j d 1.Moreover, if 1 i<j d/,d P (V i )> D P (V j ),anddd P (V i ) > DD P (V j );ifd/ i < j (d 1), D P (V i )<D P (V j ),anddd P (V i )<DD P (V j ). Proof. By the definition of D G (V) and DD G (V),wehave D P (V j )=(1++ +j)+(1++ +(d j)) = j dj+d +d, DD P (V j ) =( j )+( (d j) ) (13) = 6j +6dj 6d j 6dj + d 3 +3d +d. 6 Then D P (V i ) D P (V j ) = (i j)(i+j d), DD P (V i ) DD P (V j )= (d + 1)(i j)(i+ j d), and thus the results hold. Lemma 10. (1) For any T T 1, WW(T) WW(T n,d, d/ ), with equality if and only if T T n,d, d/. () For any T T, WW(T) WW(X n,d, d/ ),with equality if and only if T X n,d, d/. (3) For any T T 3, WW(T) WW(Y n,d, d/ ),with equality if and only if T Y n,d, d/. (4) For any T T 4, WW(T) WW(Z n,d, d/, d/ +1 ), with equality if and only if T Z n,d, d/, d/ +1. Proof. Let V = V i and two paths P = V 0 V 1 V i, Q = V i V i+1 V d inlemma6.weobtaintheresultsof(1),(),and (3). In the following, we prove (4).

4 4 Discrete Dynamics in Nature and Society Let T = Z n,d,i,j be a tree which WW(T) is as small as possible in the set T 4.LetP=V 0 V 1 V d 1 V d be a path of length d in T with d(v 0 )=d(v d )=1,and V d+1 be a pendant vertex of T which is adjacent to V j.bylemma7andthechoice of T, weknowthatj = i + 1. Note that Z n,d,i,i+1 V 0 Z n,d,i 1,i V d,andthenbycorollary5,wehave WW (Z n,d,i,i+1 ) WW(Z n,d,i 1,i ) = 3 ((n d )(i d 1) +i d+1) Note that X d+3,d, d/ Y d+3,d, d/,andthenbylemma4, we have WW (X n,d, d/ ) WW(Y n,d, d/ ) = 1 (W (X n,d, d/ ) W(Y n,d, d/ )) + 1 (S (X n,d, d/ ) S(Y n,d, d/ )) = 1 (W (X V d+3) W(Y V d+3 ) + 1 ((n d )(i d 1)(d+1) + (d+1)(i d+1)) = 1 (d+4) ((n d )(i d 1) +i d+1). (14) +D X Vd+3 (V d/ ) D Y Vd+3 (V d+1 )) + 1 (S (X V d+3) S(Y V d+3 ) +DD X Vd+3 (V d/ ) DD Y Vd+3 (V d+1 ) Thus, if i> d/, WW(Z n,d,i,i+1 ) WW(Z n,d,i 1,i )>0;if i d/, WW(Z n,d,i,i+1 ) WW(Z n,d,i 1,i ) 0, with equality if and only if d=i=n 3or d = i+1 = n.so,for any T T 4, WW(T) WW(Z n,d, d/, d/ +1 ), with equality if and only if T Z n,d, d/, d/ +1. Lemma 11. Let 3 d n 3,andthen (1) WW(Z n,d, d/, d/ +1 ) < WW(X n,d, d/ ) WW(Y n,d, d/ ); () WW(Z n,d, d/, d/ +1 )<WW(T n,d, d/ 1 ). Proof. (1) We denote Z =: Z n,d, d/, d/ +1, X =: X n,d, d/, Y=: Y n,d, d/. Note that Z n,d, d/, d/ +1 V d+1 X n,d, d/ V d+,and then by Corollary 5, we have +(D X Vd+3 (V d/ ) D Y Vd+3 (V d+1 ))) = 1 (W (X n 1,d, d/ ) W(Y n 1,d, d/ ) d+1) + 1 (S (X n 1,d, d/ ) S(Y n 1,d, d/ )+5 ( d +1) (d d )+( d +1) + 1) < 1 (W (X n 1,d, d/ ) W(Y n 1,d, d/ ) d+1) + 1 (S (X n 1,d, d/ ) S(Y n 1,d, d/ ) d+7) <WW(X n 1,d, d/ ) WW(Y n 1,d, d/ )< (17) WW (Z n,d, d/, d/ +1 ) WW(X n,d, d/ ) = 3 (D Z V d+1 (V d/ +1 ) D X Vd+ (V d+1 )) + 1 (DD Z V d+1 (V d/ +1 ) DD X Vd+ (V d+1 )) = 3 ( d + d ) + 1 (4 (d d ) (d d )) =5 4d+4 d (d d ) 5 4d+4 d 5 d<0. Namely, (15) <WW(X d+3,d, d/ ) WW(Y d+3,d, d/ )=0. Namely, WW (X n,d, d/ ) WW(Y n,d, d/ ). (18) ()ByLemma9,D Pd+1 (V d/ +1 ) D Pd+1 (V d/ 1 ),and DD Pd+1 (V d/ +1 ) DD Pd+1 (V d/ 1 ). By Corollary 5 and Lemma 9, WW (Z n,d, d/, d/ +1 ) WW(T n,d, d/ 1 ) =WW(T n 1,d, d/ ) WW(T n 1,d, d/ 1 )+ 3 D Pd+1 (V d/ +1 ) 3 D P d+1 (V d/ 1 )+ 1 DD Pd+1 (V d/ +1 ) 1 DD P d+1 (V d/ 1 )+3(n d ) WW(T n 1,d, d/ ) WW(T n 1,d, d/ 1 ) WW (Z n,d, d/, d/ +1 )<WW(X n,d, d/ ). (16) +3(n d ) = 1 (DD T n,d 1, d/ 1 (V 1 )

5 Discrete Dynamics in Nature and Society 5 DD Tn,d 1, d/ 1 (V d 1 )+3D Tn,d 1, d/ 1 (V 1 ) 3D Tn,d 1, d/ 1 (V d 1 )) + 3 (n d ) = 1 (( d ) (d d +1) +3( d (d d +1)))(n d ) +3(n d ) = 1 ( d 5d++8 d +d d ) (n d ) 1 ( d 5d++4d+d ) (n d ) = 1 ( d + )(n d ) <0. (19) Namely, WW (Z n,d, d/, d/ +1 ) <WW(T n,d, d/ 1 ). (0) Theorem 1. Let T T n,d \{T n,d, d/ }. Then WW (T) WW(Z n,d, d/, d/ +1 ), (1) with equality if and only if T Z n,d, d/, d/ +1. Proof. If d=n,notethatt n,n contains no other trees than T n,d, d/,t n,d, d/ 1,...,T n,d,,t n,d,1,andbylemma6, take V = V j and two paths P=V 0 V 1 V j, Q = V j V j+1 V d, and we have WW(T n,d, d/ ) < WW(T n,d, d/ 1 ) < < WW(T n,d, )<WW(T n,d,1 ). So, References [1] H. Wiener, Structural determination of paraffin boiling points, the American Chemical Society, vol.69,no.1,pp.17 0, [] M. Randić, Novel molecular descriptor for structure-property studies, Chemical Physics Letters, vol. 11, no. 4-5, pp , [3] D. J. Klein, I. Lukovits, and I. Gutman, On the definition of the hyper-wiener index for cycle-containing structures, Chemical Information and Computer Sciences,vol.35,no.1,pp. 50 5, [4] L. Feng and A. Ilić, Zagreb, Harary and hyper-wiener indices of graphs with a given matching number, Applied Mathematics Letters,vol.3,no.8,pp ,010. [5] L. Feng, A. Ilić, and G. Yu, The hyper-wiener index of unicyclic graphs, Utilitas Mathematica,vol.8,pp.15 5,010. [6] L. H. Feng, W. J. Liu, and K. X. Xu, The hyper-wiener index of bicyclic graphs, Utilitas Mathematica, vol. 84,pp , 011. [7] I. Gutman, Relation between hyper-wiener and Wiener index, Chemical Physics Letters,vol.364,no.3-4,pp ,00. [8] K. Xu and N. Trinajstić, Hyper-Wiener and Harary indices of graphs with cut edges, Utilitas Mathematica, vol.84,pp , 011. [9] M.-H. Liu and B. L. Liu, Trees with the seven smallest and fifteen greatest hyper-wiener indices, MATCH Communications in Mathematical and in Computer Chemistry, vol.63,no.1,pp , 010. [10] G. Yu, L. Feng, and A. Ilić, The hyper-wiener index of trees with given parameters, Ars Combinatoria, vol. 96, pp , 010. [11] H. Liu and X.-F. Pan, On the Wiener index of trees with fixed diameter, MATCH. Communications in Mathematical and in Computer Chemistry,vol.60,no.1,pp.85 94,008. WW (T) WW(T n,d, d/ 1 ), () with equality if and only if T T n,d, d/ 1 Z n,d, d/, d/ +1. If 3 d n 3,byLemmas8,10,and11,wehave WW (T) WW(Z n,d, d/, d/ +1 ), (3) with equality if and only if T Z n,d, d/, d/ +1. Competing Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments This work is jointly supported by the National Natural Science Foundation of China under Grant nos , , , and , the Natural Science Foundation of Jiangsu Province of China under Grant no. BK01741, and the Natural Science Foundation of Department of Education of Anhui Province of China under Grant nos. KJ015ZD7 and AQKJ015B010.

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