272 Acta Chim. Slov. 2005, 52, 272-281 Scientific Paper Altered Wiener Indices Damir Vukičeviča and Janez Žerovnikb " Department of Mathematics, University of Split, Croatia b Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, SI-2000 Maribor, Slovenia and IMFM, Jadranska 19, SI-1000 Ljubljana, Slovenia E-mail: janez.zerovnik@imfm.uni-lj.si Received 09-12-2004 Abstract Recently Nikohc, Irmajstić and Randić put tonvard a novel modmcation W(G) ot the Wiener number W(G), called modified Wiener index , which definition was generalized later by Gutman and the present authors. Here we study another class of modified indices defined as W^x(G)=^(y(G)xmG(u,v)x-mG(u,vfx) and show that some of the important properties of W(G), mW(G) and XW(G) are also properties of WI^!i(G), valid for most values of the parameter A. In particular, if Tn is any w-vertex tree, different from the w-vertex path Pn and the w-vertex star Sn , then for any A>1 or A < 0, Wmin!i(Pn) > ^(T,) > WI^!i(Sn). Thus for these values of the parameter A, H/millA(G) provides a novel class of structure-descriptors, suitable for modeling branching-dependent properties of organic compounds, applicable in QSPR and QSAR studies. We also demonstrate that if trees are ordered with regard to Wmin!i(G) then, in the general čase, this ordering is different for different A. Key words: Wiener number, modified Wiener indices, branching, chemical graph theory Introduction The molecular-graph-based quantity, introduced1 by Wiener in 1947, nowadays known under the name Wiener number or Wiener index, is one of the most thoroughly studied topological indices.2'3 Its chemical applications48 and mathematical properties910 are well documented. Of the several review articles on the Wiener number we mention just a few.11-13 A large number of modifications and extensions of the Wiener number was considered in the chemical literature; an extensive bibliography on this matter can be found in the reviews1415 and the recent paper.16 One of the newest such modifications was put fonvard by Nikolič, Trinajstič and Randić.17 This idea was generalized by Gutman and the present authors18 where a class of modified Wiener indices was defined, with the original Wiener number and the Nikolič-Trinajstič-Randič index as special cases. An important property of a topological index TI are the inequalities TI{Pn) > TI{Tn) > TI{Sn) or TI(Pn) < TI(Tn) < TI (Sn) (1) where Pn , Sn , and Tn denote respectively the ra-vertex path, the ra-vertex star, and any ra-vertex tree different from Pn and Sn , and n is any integer greater than 4. Such topological index may be viewed as a “branching index”, namely a topological index capable of measuring the extent of branching of the carbon-atom skeleton of molecules and capable of ordering isomers according to the extent of branching. (For more details on the problem of measuring branching see the paper19 and the references quoted therein.) Among a remarkabh/ large number of modifications and extensions of the Wiener number put fonvard recently, there are many which on trees (i.e. acyclic systems) concide12'26-31 or are linearly related with it.3237 Therefore an interesting property of a class of newly defined indices is that they provide distinct indices in the sense that they order the trees differently. More precisely, the Wiener number of a chemical graph is defined to be the sum of ali distances in the graph. W(G)= Y do{^) u,veV (o) In the papers36'38 by Gutman et al., the following modification is proposed: WX{G)= ^ dG(u,v)\ A±0 u,veV(G) It was already known to Wiener that on a tree, the Wiener number can also be computed by summing up the edge contributions, where the contribution of each edge uv is the number of vertices closer to the vertex Vukičevič and Žerovnik Altered Wiener Indices Acta Chim. Slov. 2005, 52, 272-281 273 u times the number of vertices closer to the vertex v. Formally, W(G)= J] n G (u,v)n G (v,u), (2) uveE(G) where nG(u,v) is the number of vertices closer to the vertex u than vertex v and nG(y,u) is the number of vertices closer to the vertex v than vertex u. The modified Wiener indices18 are defined as XW{G) = X nQ{u,vfnQ{v,u)x. uveE(G) Denoting n(G) = \V(G)\, the equality (2) can be also reformulated as W(G) z vsE(G) fn(G)mm{na(u,v),na(v,u)}-] mm{nG(u,v),nG(v,u)Y Let us prove this claim. Recalling that, we get W{G)= X {nG{u,v)-nG{v,u)) uveE(G) fmm{nG(u,v),nG(v,u)}-) = H54-maxK(,,v ) ,«G( v,,)}J (mm{nG{u,v),nG{v,u)}-\lX{n{G)-^{nG^)M^)})) fn(G)-mm{nG(u,v),nG(v,u)}) = uiG)[-mm{ nG ( u,v ) ,nG ( v,u )} 2 Therefore it is natural to study the following possible class of indices uveE(G) mG(u,v)Z-mG(u,v)Z) (3) which we initiate in this paper. Of course, these indices generalize the Wiener index for the trees and not for general graphs. These indices allow small modifications of the Wiener index, hence since Wiener index is of great use in the large number of QSPR and QSAR studies, these indices may improve the results obtained in such studies. For brevity, we denote: m (u,v) = min{nG (u,v),nG (v,u)} We first prove that the indices W^nX(G) for A < 0 and for A > 1 obey the inequalities (1) and can therefore be viewed as "branching indices". Theorem 1. For real number A (A > 1 or A < 0), the modified Wiener index WminX(G) satisfies the inequality where Pn , Sn , and Tn denote respectively the n-vertex path, the n-vertex star, and any n-vertex tree different from Pn and Sn , and n is any integer greater than 4. Instead of proving Theorem 1 we prove a stronger statement (Theorem 3), which may be of independent interest because it shades some light on the partial ordering induced by WrAnX(G). We also prove that the statement of Theorem 3 does not ho Id for, A e [0,1) and therefore the corresponding indices fail to properly measure branching. Furthermore, we prove that the indices studied here provide classes of distinct indices in the sense that they order the trees differently. More precisely, no matter what the values of Aj and A2 are, there always exist trees that are oppositely ordered with regard to WminXl(G) and WminX2(G). More formally, let the set of ali trees be denoted by T . Denote the set of some topological indices (e. g. the set of the modified Wiener indices Wm^x(G) for aH values of A) by 3 . We can define an equivalence relation = on the set 3 as (/2 = /2 ) <=> [ ( VT^,Tb e T ) ( i](Ta) < ^(rfc) ) » ( i2(Ta) 0 , b > 1. Hence both T' and T" havera=rai;+a+/? + l vertices. Note that the vertexr belongs to the fragmenti?. If r would be the only vertex of R , then it would be T'=T". Therefore, the only interesting čase is when nR > 2. Pt P2 P*-i Pzx-P«+i **«• 9 9 • * • • ~~i>kj qi % %-i 9b T' Pi Pi P.-i P. prio T" «M 9b X = 'lb+1 Figure 1. Graphs T' and T". Theorem 3. Let T' and T" be trees the structure of which is shown in Figure 1. Then the transformation T'-> T" increases Wmin A(G) if X> 1 and if X < 0 . First, suppose that A > 1. We shall prove that W^nX( T' )< W^nX( T"). Let T be any acyclic molecular graph with at least one edge, veV(R) and a > 0, b > 1. For the sake of simplicity, we shall denote r = p0 = q0. We have: ^mm(r')-^mm(r) = X ((n-mT.(u,V))x -mT.(u,v)2X)- L ({n-mT,{u,v))X-mT,{u,vf) = uveJT(j-) uveE(T') X [((n-mT„{u,v)Y-mr,{u,vf)-((n-mr{u,v)Y-mr{u,vf) uveE{T) + + ^ \in-mr\p. x,p.) ) -mT„( p._1,p.) )-((n-mT,( p.,p.+1 ) ) -mT,{p.,pM) J 4 + +^ l(H-/wr„(#.,#.+1)) -mT„(q.,q.+1) )-((n-mT,(q._1,q.)) -mT,(q._1,q.) J + (4) + [( (»• mT„ ( v, 9l ))* - mr„ ( v, qx)") - ( (» • mr, ( v, A ))* - mr, ( v, A fx) Note that: («?mr, (u,v)) ? -mT. (u,v) ? = (n?mr (u,v)) ? -mT (u,v) ? , for each uv eE(R), that \n'mT'' \Pi-i>Pi)) -mT'\Pi-i>Pi) ~{n'mT'\Pi>Pi+i)) -mT'\Pi>Pi+i) for each i=l,..., a, and that -mT„{qi,qi+l) =[n-mT\qi-l,qi)) -mT\qi-l,qi) for each i=l,..., b. Therefore (4) reduces to Wmn?(Tn)-Wmn?(Tf) = ((n?mr(v,ql)) ?-mr(v,ql) ? ) -((n?mT(v,p —— /7 mT,(v,qi)X (mr,(v,qi) 21\ mr,(v,A)V (M^/) 21\ « « Vukičevič and Žerovnik Altered Wiener Indices Acta Chim. Slov. 2005, 52, 272-281 275 Let fx '. —> iR be defined by fx yxj = X —X . AsA>i, f ? (x) = ?x ? -l - 2?x 2? -l = ? ? ? -l ? (l - 2x ?) > O and the limits (as x—>—) are >0 for A>1 and =0 for A=l. From definition of mG we have 2 ' < — and because of Mj,, (V, px) < /^„ (V, ^) it follows that n(G) »U? (7,") -»U? (7") = » mr.( vJ?1 ) mr(v,Pl) >0. This proves the first part of Theorem 3. When A<0, analogous reasoning gives WmnX( T") -WmnX( T' ) = n f x V v mT,( v,qi) mT,{y,pl >0, because fx \x) = X-XX-l -{\-2xX) > 0 for A < 0, concluding the proof of Theorem 3. Note that the statement Theorem 1 follows from Theorem 3, because the path Pn and the star Sn can be obtained from any tree by repeated application of the transformation T ->¦ T" or its inverse. We now show that statement Theorem 3 does not hold for other A. Lemma 4. Let Ae [0,1) and G(x,y) stand for the graph given on Figure 2. There are numbers a',a",b',b" G N such that ay+l,b>))< >Wm?(G(a",b"+l)) (6) • • • «3*^3-0 • • • • 9i v pt G(x,y) Figure 2. Graph G(x,y). Px-1 P, Note that G(x,y) has x+y+2 vertices. Similarly as in the previous section, we will compare graphs G(a,b+ 1) and G(a+ l,b). Letra = \G(a+ l,b)\ = \G(a,b+ l)\ =a+b+3. Wmin,(G{a,b+1))-Wmin,(G{a + 1,b)) = = n 2/1 ,!G(a,6+1)lv,*1J "a(a,b+1) ,«1) "G(a+1,b) ,P1) mG(a+1,b){v,P1) 2/1"! 2X\ (a + b + 3f a + 2 Y f a + 2 T) ' a + b ^j [^b^j a+1 Y 22 N J From the L agrange's the orem, it follow s that there are num bers r,se (0,1) such that '', mm,/l (G(a,* + l))-^mm/l(G(a + l,*)) = (a + * + 3)21 /L v -2/1 a __________ (7) Vukičevič and Žerovnik Altered Wiener Indices n n n n 276 -------------------- Therefore, — Acta Chim. Slov. 2005, 52, 272-281 ^( GM + l))-^,,( G ( a+l,»)) > ( a+»+3 ) a + o + 3 -/L ----------- • 1-2---------'—t-.------------ V ; U + b + 3) (a + \f-1 (a + b 22 ( a + \ ] 22-1 Note that lim (tf + l) ( a + 6 + 3 )" ' _ ( a + 2 ) 21-1 * i ^ (a + lf1 (a + b+ 3)" Therefore, if we take a'= a and sufficientlvla rge b' = b'(a') we have W x{G( a:V+l)) -W^(G(*+l,»)) >0 ,Tmm,/lVW mm,2 This proves (5). Let us return to the relation (7). We have WmmA(G(a,b + l))-WmmA(G(a + l,b)) = (a + b + 3) It followsthat 22 Wmm^G ( a,b + l))-Wmm^G( a + l,b )) <{a + b + 3) For b= a + 1 the last ineaualitv reads f J V f M a + l + r a + b + 3 2-1 ^J _u(a + l + S 22 f a + 2 l^+T+7 2-1 WmmA(G{a,a + 2))-WmmA(G{a + l,a + l))<(2a + 4f-X Note that lim 2a + 4 2-1 f a + 2 Ua + 4 2/1 -2 a + 1 22-A y 22-1 a + b + 3 / 1 a + 2 V f a + 1 -2 x22-A 2a + 4 \2-l /- x22-l / \M 2 "2-2 =2 1-2 \Ja+4 \ 2-1 - •[l-21"^]<0. Therefore, for the sufficiently large a" and b"= a" + l, we have WmmX [G(a'\b"+l)j —Wmmx \GyCt"+l,b")) < 0. This proves the relation (6) and Lemma 4. Proof of Theorem 2 Let us define - P{a,b,c,r]) " -2V + (a + *)-([(a + 3* + 1)-1]"-1") + *-([(a + 3* + 1)-2]"-2'') +b-([(a + 3b + 1)-3j -3t>)-(a + 3b-c)-([(a + 3b + 1)-1J -1«}- [{a + 3b + 1)-2j -2») Q(a,b,c,?j) v-2"+ (a + b)-([(a + 4b + 1)-1f -1") + b-([(a + 4b + 1)-2]' -2") +b-([{a + 4b + 1y ¦3]-3") + b-([{a + 4b + 1y -4]-4") (a + 4b-c)-([(a + 4b + 1y ¦1]-1)-c-([(a + 4b + 1)-2]' -2") Vukičevič and Žerovnik Altered JViener Indices Acta Chim. Slov. 2005, 52, 272–281 277 Furthermore, observe that for the graphs G' (a,b) and//' (a,b,c) with a+3b + l vertices sketched on Figure 3, we get, after straighfonvard computation using only the definition of WminX, W min,? (G'(a,b))-Wmmv(H'(a,b,c))=P(a,b,c,l]) C b G'(a,b) H\a,b,c) Figure 3. Graphs G' (a,b) and//' (a,b,č). Denote by G" (a,b) and H" (a,b,c) graphs on a+Ab+ 1 vertices on Figure 4. C G"(a,b) b Figure 4. Graphs G” (a,b) and H” (a,b,c). H"(a,b,c) . We will now prove that for each pair of distinct A and tj there are numbers a, b, c such that at least one pair of corresponding graphs is ordered differenth/ by the corresponding indices. Distinguish three cases: ČASE 1: (A > 0 and rj < 0) or (A < 0 and rj > 0). Without loss of generality, we may assume that A > 0 and r\ < 0. Note that 2A +1^2''+1 or equivalently that 2A+3A + 41-3 2A + 3A-2 2" + 3" + 4" - 3 2"+3"-2 2*-1 2~1 At least one of the following subcases applies: * 2"-1 2"-1 Vukičevič and Žerovnik Altered Wiener Indices 278 Acta Chim. Slov. 2005, 52, 272-281 2*+3*-2 2"+3"-2 SUBCASE 2.1:----------------^----------------. 2x-1 2"-1 T ¦ 2x+3x-2 2"+3"-2 2^+3^-2 2"+3"-2 lwo subsubcases, can be observed, l.e. ----------------<---------------- and ---------------->---------------- . 2x-1 2"-1 2x-1 2"-1 2^+3^-2 2"+3"-2 Since thev are solved in similar way, we shall assume that----------------<---------------- . Hence, there is a rational 2x-1 2"-1 2^+3^-2 2"+3"-2 c numbera such that----------------< a <----------------. Denote a = — c & e A/ and let us calculate 2x-1 H 2"-1 H b \[mPiabc>Al=\[m(( a + b )-\x+b-2x + b-3x- ( a + 3b-c )-\x-c-2x) = a^(a + 3b + l) a^v = b ( 2x+3x-2 ) -c-( 2x-l) = b-( 2x-\y( 2x-\ P(a,b,c,T}) Note that lim---------------------------------------------------------------------------------------- = 1 Hence, for sufficiently large a, the last expression is positive. Let us calculate the denominator of this expression: >0 v iV 2"+3"-2 b(2-2" -3") + (2" -1)c = b(2" -A2'2" ~3V +q] = b(1-2") 2"-1 Therefore, there is a sufficiently large a e N such that P(a,b,c,X) < 0 and P(a,b,c,tj) > 0 and hence, for the graphs G' (a,b) and H' (a,b,c) graphs depicted on Figure 3, W^x(Gia,b))-W^x(Hia,b,c)) = P(a,b,c,X)<0; Wm^(G>{a,b))-Wmmv(H>(a,b,c)) = P(a,b,c,71)>0 The claim is proved in this subcase. 2^+3^ + 4^-3 2"+3"+4"-3 SUBCASE 2.2: ----------:-------------^-----------------------. 2*-1 2"-1 2i+3i +4^-3 2"+3"+4"-3 2^+3^+4^-3 2" + 3" + 4" -3 Again, there are two subsubcases:-----------------------<-----------------------and-----------------------> 2x-1 2"-1 2x-1 2"-1 21+31+41-3 2"+3"+4"-3 Since both cases can be treated analogouslv, we assume that ----------------------<---------------------- . Hence, 2*-1 2"-1 2^+3^ + 4^-3 2"+3"+4"-3 c there is a rational number g such that-----------------------~= ^^^ = lim( ( a + *)-l1 + *-21+*-31+*-41- ( a + 4*-c )-l1-c-21) b? ( 2?+3?+4?-1 ) -c ( 2?-1 ) = b ( 2?-1 )( 2? + 3? + 4?-1 + 3?+4?-1)-c(2?-1) = 6(2?-1)| ? - ------q Vukičevič and Žerovnik Altered Wiener Indices <0 Acta Chim. Slov. 2005, 52, 272-281 279 b(3-2"-3"-4")-bq(\-2'') = b(\-2'') Q(a,b,c,J]) On the other hand, Hm 7=----------------------------------------------------------------------------------------------=r = l — [(a + ž,)(-l") + ž,.(-2'') + 6(-3'') + 6(^)-(a + 4*-C)(-l'')-C(-2'')] Hence, for sufficiently large the last expression is positive. Let us calculate the denominator of this expression: (a + b)(-Vi) + b.(-2«) + b(-3«) + b(-4«)-(a + 4b-c)(-V>)-c(-2«) = f2"+3"+4"-3 ^ ------------------------a >0. ^ 2"-l H) Therefore, there is flLW such that Q(a,b,c,JJ~\ > 0 and G(a,b,c,7]J < 0, which implies, for the graphs G"[a,b)and H"(a,b,c) given on Figure 4, ^mb,A(G;"( a,6)) -»r^(^"( a,6,c )) = e(a,ft,c,A)<0 Wm^(G"{a,b))-Wmmti(H"(a,b,c)) = Q{a,b,c,7])>0. ČASE 2: X,jU>0- Note that 2 +1^2'+1 or equivalently that 2^ + 3A + 4A-3 2A + 3A-2 2" + 3" +4" -3 2^+3^-2 2*^1 2^T* 2^1 2^T~ At least one of the following must hold: 2^+3^-2 2?+3?-2 SUBCASE 2.1: ------5---------^? . 2*-1 2*-1 2*+3*-2 2?+3?-2 Without loss of generahty, we may assume that ----------------< ? Hence, there is a rational number 2X-1 2V -1 2*+3*-2 2?+3?-2 c q such that----------------< a < ? Denote a = —,c,bG N . Let us calculate 2x-\ 271 -1 b P(a,b,c,A) ,¦/, AVli A 0i i~x / ~, 1 ^ f21 + 31-2 "I ft(2^+3^-2)-c-(2^-i)=ft-(2^-i). ::: ~q <0 P(a,b,c,u) Completelv analogouslv, we get lim------------------> 0. Lherefore, there is a sufficentlv large aeW such that a^(a + 3b + lf for graphs G' (a,b) and H' (a,b,c) from Figure 3. Lhe claim is proved in this subcase. 2^+3^+4^-3 2?+3?+4?-3 SUBCASE 2.2:-----------------------^ ? . 2*-1 2^-1 2x+3x+4x-3 2?+3?+4?-3 Without loss of generahtv, we may assume that ----------------------- < ? Hence, there is a rational 2X-1 2^-1 2x+3x+4x-3 2?+3?+4?-3 c , Y, number q such that -----------------------< a < ? Denote q = —, o, C ? W and Vukičevič and Žerovnik Altered Wiener Indices 280 Acta Chim. Slov. 2005, 52, 272–281 lim Q(a,b,c,X) °*(a + 4b + iy Xim{{a + b)-lx+b-2x + b-3x + b-4x-{a + 4b-c)-lx-c-2x) b?2 ?+3? + 4?-1 ) -c ( 2?-1 ) = b ( 2?-1 )( 2 ? -1 <0 Q(a,b,c,H) A whereas lim------------------> O. -" (a+ 46 + 1)" Therefore, there is a sufficiently large a G Wsuch that, for graphs G"{a,b) and H"ya,b,c) (see Figure 4), ^f7(G»(a>6))-^minj'f/(^»(a>6>C)) = G(a>6>C^)>0. ČASE 3: A,J]< 0 . This čase can be solved by a similar techniques to ones used in the proof of the Čase 2. We omit the details. Conclusions The Wiener number is one of the most useful indi-ces in the QSPR and QSAR studies. It is well correlated with a number of physical and chemical properties of chemical compounds. Therefore it is natural to study the indices that represent small alternations of this index that may give better correlations with specific properties than the original Wiener index. In this paper the family of such alternations is presented as topological index W^nX (where H/min4 is original Wiener index. The farnih/ of these indices generalizes the notion of the Wiener index and may improve results of QSPR and QSAR studies. However, this generalization is valid only for trees and can not be applied to cyclic graphs. On the other hand, the indices studied here can be applied on weighted graphs. A natural generalization to weighted graph would be obtained by replacing the numbers nG(u,v) and n(G)= \V(G)\ with the sums of weights of the corresponding vertices. Acknowledgements This work was in part supported by the Ministry of Science of the Republic Croatia and by the Ministry of Education, Science and Šport of Slovenia. The authors wish to thank the anonymous referees for constructive remarks. Special thanks are due to the journal editors for their effort in proofreading and reformatting the manuscript. References 1. H. Wiener,/. Am. Chem. 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Chem 1997, 36A, 128-132. Povzetek Nedavno so Nikolič, Trinajstič in Randič predlagali modifikacijo Wienerjevega števila W(G), definirano z mW(G)=*Ž nG(u,v)~l nG(u,v)~l.Invarianto so Gutman in avtorja posplošili naxW(G)= J nG(u,v)xnG(u,v)x. Tu obravnavamo posplošitev podobnega tipa, W- A(G)= 2, (V(G) mG(u,v) -mG(u,v) ) in pokažemo, da nekatere uveE(G) pomembne lastnosti M^G), W(G) and W(G), veljajo tudi za vr^^fG), za večino vrednosti parametra A. Dokažemo, da za poljubno drevo (povezan acikličen graf) z n točkami Tn, ki ni pot Pn ali zvezda Sn , velja ^nim,x(Pn) > ^mm.iijtd > ^minA^n)'za vse A > 1 in A < 0. Za te vrednosti parametra je torej Wm-m t(G) razred topoloških indeksov, ki so lahko uporabni pri obravnavi od razvejanosti odvisnih lastnosti v QSPR in QSAR. Dokažemo tudi, da so vsi novi indeksi različni v naslednjem smislu: če uredimo vsa drevesa glede na WmhiX(G) potem za različne vrednosti parametra A dobimo različne urejenosti. Vukičević and Žerovnik Altered Wiener Indices