THE TOP TEN VALUES OF HARMONIC INDEX IN CHEMICAL TREES

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1 Kragjevac J. Sci. 37 (25) UDC 54.27:54.6 THE TOP TEN VALUES OF HARMONIC INDEX IN CHEMICAL TREES Aliye Zolfi, Ali Rea Ashrafi a Siros Morai 2 Departmet of Mathematics, Faclty of Mathematical Scieces, Uiversity of Kasha, Kasha , I. R. Ira 2 Departmet of Mathematics, Arak Uiversity, Arak, I. R. Ira s: ashrafi_385@yahoo.co.i, ashrafi@kasha.ac.ir (Receive Jly 3, 24) ABSTRACT. Let G be a verte graph with egree seqece, 2,,. The harmoic ie H(G) is efie as / I( G), where I(G) = ( / ). I this paper the top i= te vales of harmoic ie i the set of all chemical trees of orer are etermie. Keywors: Harmoic ie, chemical tree. i INTRODUCTION We se West [] for termiology a otatio ot efie here a cosier fiite simple coecte graphs oly. Sppose G is sch a graph with V(G) = {v, v 2,, v }. If we sort vertices of G i sch a way that eg(v ) eg(v 2 ) eg(v ) the the seqece (, 2,, ) is calle a egree seqece for G, where i = eg(v i ), i. A graph ivariat is ay fctio o a graph that oes ot epe o a labelig of its vertices. A big mber of ifferet ivariats have bee employe to ate i chemistry for solvig some chemical problems. Here we are itereste to the harmoic ie efie as H(B) = /I(B), where I(B) = i = ( / i ), for a graph B. This topological ie was itroce by Narmi [2]. A chemical tree is a tree i which every verte has egree at most 4. We eote by g(), the set of all verte chemical trees. It is easy to see that if A a B are two elemets of g() with the same egree seqece the H(A) = H(B). This is motivatio for efiig a eqivalece relatio ~ o g() by A ~ B if a oly if A a B have the same egree seqece. Sppose () eotes the set of all eqivalece classes of ~ o () a T, T 2 (). Defie T T2 if a oly if for each elemet A T a B T2, we have H(A) H(B). The aim of this paper is to compte the first maimm vale of harmoic ie. We ecorage the reaer to coslt [3 8] for basic comptatioal techiqes o the problem.

2 92 MAIN RESULTS I this sectio, we are aalyig chemical trees with k th, k, maimm vales for the harmoic ie. I orer to formlate or reslts, we ee itroce some graph otatios se i this paper. Defie: T,H () = {M CT() T CT() ; H(T) H(M)}, a for each i, i r = CT(), we have: T i,h () = {M CT() T T i } T CT() T T i ; H(T) H(M)}, where r is the mber of verte chemical trees. The elemets of T i,h () are calle i th maimm class of chemical trees with respect to H ie. Lemma. Let T be a verte chemical tree a T ' is a verte chemical tree obtaie from T by eletig a peat verte a appeig a peat verte to aother peat verte of T. The I( T ) I(T ). ' Proof. Sppose v is a peat ege of T, eg(v) =, eg() 2 a w is a peat verte of T sch that T is obtaie by eletig v a appeig it to w. The ' eg T (w) = = eg ' (w), eg ( ) eg ( ) T T = T ' + a for aother verte ifferet from a w, eg T () = eg ' (). O the other ha, I ( T ) T =, w + + = w, w + + a I (T =, w + + ') = w +, w + +. Ths + + which is eqivalet to, 2 2 ( ) 2 provig the lemma. Corollary 2. Let T be a verte chemical tree the I(T) + /2 with eqality if a oly if T is a path of legth. The seco miimm vale of I ie is 4/3 + /2 a it is attaie if a oly if T is isomorphic to P, where P is a tree with eactly three peat vertices. Lemma 3. Let T be a verte chemical tree cotaiig vertices, sch that eg T () = 2 a eg T () = 2 or 3. Sppose T a T 2 are maimal sbtrees of T cotaiig as a peat verte a V(T 2 ), Figre. If T is the chemical tree costrcte from T a T 2 by ietifyig a the I(T ) I( T ), Figre 2. Proof. It is easy to see that eg T () = 2, eg () =, eg () eg () T = T + a for T aother arbitrary verte, eg T () = eg (). T Figre. The Chemical Tree T Cotaiig a Sbtree T.

3 93 Figre 2. The Chemical Tree T Costrcte from T a T 2. Therefore, I ( T ) =, + + =, I ( T ) =, + +. Sice + < +, < <, provig the lemma. ( + ) 2 a a so Lemma 4. Let T be a verte chemical tree cotaiig vertices a of egree 3. We also assme that T, T 2 a T 3 are maimal sbtrees of T with as a peat a T 3. If " T is the chemical tree costrcte from T a T 2 by ietifyig a the I(T ) I( T ). Proof. By or assmptio eg T () = 3, eg " () = 2, eg () 3 T T = a eg " () = 4. T Ths, I ( T ) =, + + = + +,, 3 3 " I ( T ) =, + +, 2 4 which implies that I(T ) I( T ). " Figre 3. The Chemical Tree T. Figre 4. The Chemical Tree T. Remark 5. Sppose T is a chemical tree with maimm I ie a r(i), i 4, eotes the mber of vertices of egree i. The by Lemmas 3 a 4, r(2) + r(3) =,.

4 94 I what follows g() eotes the set of all verte chemical trees. Corollary 6. A verte chemical tree T has the maimm vale of I ie i g(), if T has the maimm mber of peats i g() a o peat vertices satisfyig r(2) + r(3). I Table, the mber of peats, the mber of vertices of egree 2, the mber of vertices of egree 3 a the mber of vertices of egree 4 are compte for the maimal chemical trees with respect to I ie, whe the mber of vertices are at most 6. I Table 2, the verte chemical trees with respect to I ie, 4 3, are epicte. Table. The Nmber of Vertices of Each Degree i Verte Chemical Trees, 4 6. # Vertices # Vertices # Vertices of # N of Degree 2 of Degree 3 Degree 4 Peats 3 4= = = = = = = = = = = = =3 5+ Table 2. The Maimal Chemical Trees with respect to I Ie, 4 3. =4 =5 =6 =7 =8 =9 = = =2 =3

5 From Table 2, it is atral to ask whether or ot the maimal tree with respect to I ie is iqe. I the followig eample we respo egatively to this qestio. Eample 7. Cosier the graphs A a B epicte i Figres 5 a 6, respectively. By simple calclatios, oe ca see that A a B are o-isomorphic graphs with the same I ie. This shows that the maimm of I ie ca be occrre i more tha 2 chemical trees. 95 Figre 5. The Graph A. Figre 6. The Graph B. Sppose g() eotes the set of all verte chemical trees, = 3k + t, t 3, 3 a T g() is a verte chemical tree havig maimm I ie. Oe ca easily see that r() = 2k + 2 /t a r(2) + r(3) =. If t = 3 the r(2) =, r(3) =, a if t = the r(2) = a r(3) =. Therefore, r(4) = r() r(2) r(3) = (2k + 2 /t ) (r(2) + r(3)) = 3k + r 2k 2 + /t = k + t 3 + /t. Remark 8. Sppose T g(), = 3k + t, t 3, a 3. The the maimm of I ie is occrre if r() = 2k + 2 /t, r(2) + r(3) = a r(4) = k + t 3 + /t. I this case, the vale of I ie is compte by I(T) = 2k + 2 /t + ¼(k + r 3 + /r ) + r(2)/2 + r(3)/3. Moreover, if = 3k + 3 the r(2) =, r(3) = a if = 3k + the r(2) = a r(3) =. Corollary 9. Sppose T is a verte chemical tree havig maimal I ie i g(), 4. Costrct the chemical tree S from T by eletig a peat coecte to verte v of egree 4 i T a coectig to a verte of egree 2. If there is ot a verte of egree 2 the we coect it to a peat. The the chemical tree S attai the seco maimm vale of I ie. Eample. I this eample two chemical trees C a D are costrcte sch that the mber of peats of C is greater tha D, bt I(D) < I(C). These are epicte i Figres 7 a 8. It is easy to calclate I(D) = 7 + /4 < 7 + /6 = I(C). Figre 7. The Graph C. Figre 8. The Graph D.

6 96 We ow compte the first te vales of harmoic ie i the class of all chemical trees. At first, it is a easy fact that the path P a the chemical tree P have the maimm a seco maimm harmoic ie, respectively. Figre 9. The Graph P. To fi the thir maimm vale of harmoic ie, we cosier two classes X a Y of chemical trees with eactly for peat vertices i which X has eactly two vertices of egree 3 a remaiig vertices of egree 2, a Y has eactly oe verte of egree 4 a remaiig vertices has egree 2. Remark. The thir a forth maimm vales of harmoic ie i the class of chemical trees will attai i X a Y, respectively. We are ow reay to compte the fifth a sith vales of harmoic ie i the class of chemical trees. We cosier verte chemical trees with five peat vertices. These are i the form of graphs which are epicte i Figres or. These vales for the fifth a sith maimm vales of verte chemical trees, 8, are 2/(+4) a 2/(6+25), respectively. Figre. Figre. Figre 2. Figre 3.

7 97 Figre 4. Figre 5. Apply agai or algorithm to compte the seveth, eighth a ith maimm vales i the class of all verte chemical trees. These chemical trees have eactly si peats a ca be raw as Figres 2 4. The teth maimm vale ca be occrre i chemical trees havig seve peats. This chemical tree is epicte i Figre 5. We recor or mai reslt i the followig theorem: Theorem 2. For each δ, δ, Tδ, H () = {Aδ, }, where chemical trees A δ, are epicte i Table 3. Ackowlegemet The research of the first a seco athors are partially spporte by the Uiversity of Kasha er grat o 592/84. Refereces: [] D. B. WEST, Itroctio to Graph Theory, Pretice Hall, Ic., Upper Sale River, NJ, 996. [2] H. NARUMI, New topological iices for fiite a ifiite systems, MATCH Comm. Math. Compt. Chem. 22 (987) [3] I. GUTMAN a M. GHORBANI, Some properties of the Narmi Katayama ie, Appl. Math. Lett. 25 (22) [4] I. GUTMAN a H. NARUMI, A ieqality for the simple topological ie, Coll. Sci. Pap. Fac. Sci. Kragjevac (99) [5] I. GUTMAN, A property of the simple topological ie, MATCH Comm. Math. Compt. Chem. 25 (99) 3 4. [6] M. FISCHERMANN, I. GUTMAN, A. HOFFMANN, D. RAUTENBACH, D. VIDOVIĆ a L. VOLKMANN, Etremal Chemical Trees, Z. Natrforsch. 57 a (22) [7] M. J. NADJAFI-ARANI, G. H. FATH-TABAR a A. R. ASHRAFI, Etremal graphs with respect to the verte PI ie, Appl. Math. Lett., 22 (29) [8] A. ZOLFI a A. R. ASHRAFI, Etremal properties of Narmi Katayama ie of chemical trees, Kragjevac J. Sci. 35 (23) 7 76.

8 98 Table 3. The k th Maimm of Harmoic Ie of Chemical Trees, 3 k. Names K #Vertices H Ie The k th Maimm of Harmoic Ie A 3,5 k = 3 = 5 2/7 A 3, k = 3 6 6/(3+) A 4, k = 4 6 4/(2+7) A 5,7 k = 5 = 7 84/67 A 5, k = 5 8 2/(+4) A 6, k = 6 8 2/(6+25) A 7,8 k = 7 = 8 6/3 A 7,9 k = 7 = 9 8/83 A 7, k = 7 6/(3+4) A 8,9 k = 8 = 9 9/7 A 8, k = 8 2/(6+29) A 9, k = 9 2/(+5) A, k = = 6/47 A, k = = 4/3 A, k = 2 6/(3+6)

A probabilistic proof of a binomial identity

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