COMPUTING SZEGED AND SCHULTZ INDICES OF HAC C C [ p, q ] NANOTUBE BY GAP PROGRAM

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1 Diges Journal of Nanomaerials and Biosrucures, Vol. 4, No. 1, March 009, p COMPUTING SZEGED AND SCHULTZ INDICES OF HAC C C [ p, q ] NANOTUBE BY GAP PROGRAM A. Iranmanesh *, Y. Alizadeh Deparmen of Mahemaics, Tarbia Modares Universiy P. O. Box: , Tehran, Iran In his research, we give a GAP program for compuing he Szeged and Schulz indices of any graph. Also we compue he Szeged and Schulz indices of HAC C C [, ] p q by his program. (Received January ; acceped February 1, 009) eywords: Szeged index, Schulz index, Nanoube, GAP programming. 1. Inroducion Le G be a conneced graph. The verex-se and edge-se of G denoed by V(G) and E(G) respecively. The disance beween he verices u and v, d(u,v), in a graph is he number of edges in a shores pah connecing hem. Two graph verices are adjacen if hey are joined by a graph edge. The degree of a verex i V ( G) is he number of verices joining o i and denoed by v() i. The (, i j) enry of he adjacency marix of G is denoed by Ai (, j ). Le e be an edge of a graph G connecing he verices u and v. Define wo ses N ( e G ) and N ( ) 1 e G as follows: N ( e G) = { x V ( G) d( x, u) < d( x, v)} and 1 N ( e G) = { x V ( G) d( x, v) < d( x, u)}. The number of elemens of N ( e G ) and N ( e G ) are denoed by n ( e G ) and 1 1 n ( e G ) respecively. A opological index was inroduced by Guman and called he Szeged index, abbreviaed as Sz [1] and defined as Sz ( G) = n1( e G). n( e G). e E ( G) Schulz index (MTI) is a opological index was inroduced by Schulz in 1989, as he molecular opological index [], and i is defined by: MTI = v ()( i d (, i j) + A(, i j)). { i, j} V ( G) In his paper, we give an algorihm ha enables us o compue he Szeged and Schulz indices of any graph. Also by his algorihm, we compue he Szeged and Schulz indices of HAC5C [, ] 6C7 p q nanoube. * Corresponding address: iranmana@modares.ac.ir

2 68. Resul In his secion, we give an algorihm ha enables us o compue he Szeged and Schulz indices of any graph. For his purpose, he following algorihm is presened: 1- We assign o any verex one number. - We deermine all of adjacen verices se of he verex i, i V (G) and his se denoed by N(i ). The se of verices ha heir disance o verex i is equal o ( 0) is denoed by D i, and consider D i, 0 = { i}. Le e = ij be an edge connecing he verices i and j, hen we have he following resul: a) V = U 0 Di,, i V (G). b) di (, j) = Di,, i VG ( ) j V ( G) 1 c) MTI ( G) = v ( i ) ( d ( i, j ) + A( i, j)) = v ( i ) + d ( i, j ) i V ( G) j V ( G) i V ( G) j N ( i) j V ( G)\ N ( i) = v( i) + v( i) Di, i V ( G) j Di, b) ( Di, \ D j, ) ( D j, 1 U D j, + 1), 1. c) ( Di, ID j, 1) N ( e G ) and Di, I D j, + 1 N 1( e G ) 1. d) ( Di, 1 U { i}) \ ( D j,1 U{ j}) N1( e G) and ( D j,1 U{ j}) \ ( Di,1 U{ i}) N ( e G). According o he above relaions, by deermining D i,, 1, we can obain N 1( e G) and N ( e G) for each edge e and herefore he Szeged and Schulz indices of he graph G is compued. In he coninue we obain he D i,, 1, for each verex i. 3- The disance beween verex i and is adjacen verices is equal o 1, herefore D i, 1 = N(i ). For each j Di,, 1, he disance beween each verex of se N j) \ ( D i U D ) and he verex i is equal o + 1, hus we have (, i, 1 Di, + 1 = U j D ( N( j) \ ( D, D, 1), 1., i U i i D, According o he above equaion we can obain i for each i V (G). 4- In he sar of program we se SZ and Sc equal o zero and T equal o empy se. In he end of program he values SZ and Sc are equal o he Szeged and Schulz indices of he graph G respecively. For each verex i, 1 i n, and each verex j in N (i ), we deermine N1( e G) and N ( e G) for edge e = ij, hen add he values of n 1( e G). n ( e G) o S Z. Since he edge j i is equal o ij, we add he verex i o T and coninue his sep for he verex i+1 and for each verex in N ( i + 1) \ T. GAP sands for Groups, Algorihms and Programming [3]. The name was chosen o reflec he aim of he sysem, which is group heoreical sofware for solving compuaional problems in group heory. The las years have seen a rapid spread of ineres in he undersanding, design and even implemenaion of group heoreical algorihms. GAP sofware was consruced by GAP's eam in Aachen. We encourage he reader o consul Refs. [4] and [5] for background maerials and compuaional echniques relaed o applicaions of GAP in solving some problems in chemisry and biology. The molecular opological index sudied in many papers [6-9]. In a series of papers, he Szeged and Schulz indices of some nanoubes compued [10-13], anoher opological indices are compued [14-18]. 68

3 3. Disscussion and conclusions 69 A C 5C6C7 ne is a rivalen decoraion made by alernaingc 5, C 6 andc 7. I can cover eiher a cylinder or a orus. In his secion we compue he Szeged and Schulz indices of HAC 5C6C7 nanoube by GAP program. Fig. 1. HAC 5C6C7[4,] Nanoube We denoe he number of penagons in he firs row by p. In his nanoube he hree firs rows of verices and edges are repeaed alernaively; we denoe he number of his repeiion by q. In each period, here are 16 p verices and p verices are joined o he end of he graph and hence he number of verices in his nanoube is equal o16 pq + p. We pariion he verices of his graph o following ses: 1: The verices of firs row whose number is5 p. : The verices of he firs row in each period excep he firs one whose number is 5 p ( q 1). : The verices of he second rows in each period whose number is 6 pq. 3 4 : The verices of he hird row in each period whose number is5 pq. 5 : The las verices of he graph whose number is p. We wrie a program o obain he adjacen verices se o each verex in he ses i, i=1 5. We can obain he adjacen verices se o each verex by he join of hese programs. The following program compues he Szeged and Schulz indices of HAC [ 5C6C7 p, q] nanoube for arbirary p and q. p:=9; q:=9;#(for example) n:=16*p*q+*p; N:=[]; 1:=[1..5*p]; V1:=[..5*p-1]; for i in V1 do if i mod 5 =1 hen N[i]:=[i-1,i+1,(i-1)*(6/5)+1+5*p]; elif (i mod 5) in [0,] hen N[i]:=[i-1,i+1]; elif i mod 5=3 hen N[i]:=[i-1,i+1,(i-3)*(6/5) +3+5*p]; elif i mod 5=4 hen N[i]:=[i-1,i+1,(i-4)*(6/5)+5+5*p]; fi; od; N[1]:=[,5*p,5*p+1]; N[5*p]:=[1,5*p-1]; k:=[5*p+1..16*p*q]; k:=filered(k,i->i mod (16*p) in [1..5*p]); 69

4 70 for i in k do x:=i mod (16*p); if x mod 5=1 hen N[i]:=[i-1,i+1,(x-1)*(6/5) +1+i-x+5*p]; elif x mod 5= hen N[i]:=[i-1,i+1,i-5*p+1]; elif x mod 5=3 hen N[i]:=[i-1,i+1,(x-3)*(6/5)+3+i-x+5*p]; elif x mod 5=4 hen N[i]:=[i-1,i+1,(x-4)*(6/5)+5+i-x+5*p]; elif x mod 5=0 hen N[i]:=[i-1,i+1,i-5*p];fi; if x=1 hen N[i]:=[i+1,i-1+5*p,i+5*p];fi; if x=5*p hen N[i]:=[i-1,i-5*p,i-5*p+1];fi; od; k3:=filered(k,i->i mod(16*p) in [5*p+1..11*p]); for i in k3 do x:= (i-5*p) mod (16*p); if x mod 6=1 hen N[i]:=[i-1,i+1,(x-1)*(5/6)+ i-x-5*p+1]; elif x mod 6= hen N[i]:=[i-1,i+1,(x-)*(5/6)++i-x+6*p]; elif x mod 6=3 hen N[i]:=[i-1,i+1,(x-3)*(5/6)+3+i-x-5*p]; elif x mod 6=4 hen N[i]:=[i-1,i+1,(x-4)*(5/6)+4+i-x+6*p]; elif x mod 6=5 hen N[i]:=[i-1,i+1,(x-5)*(5/6)+4+i-x-5*p]; elif x mod 6=0 hen N[i]:=[i-1,i+1,x*(5/6)+1+i-x+6*p];fi; if x=1 hen N[i]:=[i+1,i+6*p-1,i-5*p];fi; if x=6*p hen N[i]:=[i-1,i+1,i-6*p+1];fi; od; k4:=filered(k,i-> i mod (16*p) in Union([11*p+1..16*p-1],[0])); for i in k4 do x:=(i-11*p) mod (16*p); if x mod 5 =1 hen N[i]:=[i-1,i+1,(x-1)*(6/5)+i-x-6*p]; elif x mod 5 = hen N[i]:=[i-1,i+1,(x-)*(6/5)++i-x-6*p]; elif x mod 5 =3 hen N[i]:=[i-1,i+1,i-1+5*p]; elif x mod 5 =4 hen N[i]:=[i-1,i+1,(x-4)*(6/5)+4+i-x-6*p]; elif x mod 5 =0 hen N[i]:=[i-1,i+1,i+5*p];fi; if x=1 hen N[i]:=[i-1,i+1,i-1+5*p];fi; if x=5*p hen N[i]:=[i-1,i-5*p+1,i+5*p];fi; od; 5:=[16*p*q+1..n]; for i in 5 do x:=i-16*p*q; if x mod =0 hen y:=(5/)*x +16*p*q -5*p; else y:=(5/ )*(x-1) +3+16*p*q -5*p; fi; N[i]:=[y]; N[y][3]:=i; od; Sc:=0; v:=[]; D:=[]; for i in [1..n] do D[i]:=[]; u:=[i]; D[i][1]:=N[i]; v[i]:=size(n[i]); u:=union(u,d[i][1]); Sc:=Sc+v[i]**Size(D[i][1]); r:=1; :=1; while r<>0 do D[i][+1]:=[]; for j in D[i][] do for m in Difference (N[j],u) do AddSe(D[i][+1],m); od; od; u:=union(u,d[i][+1]); Sc:=Sc+v[i]*(+1)*Size(D[i][+1]); if D[i][+1]=[] hen r:=0;fi; :=+1; od; od; T:=[]; sz:=0; for i in [1..n-1] do 70

5 N1:=[]; for j in Difference(N[i],T) do N:=[]; N1[j]:=Union(Difference(N[i],Union([j],N[j])),[i]); N[i]:=Union(Difference(N[j],Union([i],N[i])),[j]); for in [..Size(D[i])-1] do for x in Difference(D[i][],Union(D[j][],[j])) do if no x in D[j][-1] hen AddSe(N1[j],x); elif x in D[j][-1] hen AddSe(N[i],x); fi; od; od; sz:=sz+size(n1[j])*size(n[i]); od; Add(T,i); od; sz; # (The value of sz is equal o Szeged index of he graph) Sc; # (The value of Sc is equal o Schulz index of he graph) 71 Table1. Szeged and Schulz indices of HAC C C p, ] nanoube [ q p q Szeged Schulz References [1] P.P. hadikar, N.V. Deshpande, P.P. ale, A.A. Dobrynin, I. Guman, G. Domoor, J. Chem. Inf. Compu. Sci. 35, 545 (1995). [] H.P. Schulz. J. Chem. Inf. Compu. Sci. 9, 7 (1989). [3] M. Schoner, e al., GAP, Groups, Lehrsuhl De fur Mahemaik, RWTH: Achen, (199). [4] M. Dabirian, A. Iranmanesh. MATCH Commun. Mah. Compu. Chem. 54, 75 (005), [5] N. Trinajsic, CRC Press: Boca Roon, FL, (199) [6] H.P. Schulz, J. Chem. Inf. Compu. Sci. 40, 1158 (000). [7] S. lavžar, I. Guman. Disc. Appl. Mah. 80, 73 (1997). [8] I. Guman. J. Chem. Inf. Compu. Sci. 34, 1087 (1994). [9] A.A. Dobrynin. Croa. Chem. Aca. 7, 869 (1999). [10] A. Iranmanesh, Y. Alizadeh. Am. J. Applied Sci., 5, 1754 (008). [11] A. Iranmanesh, Y. Alizadeh. Inernaional Journal of Molecular Sciences, 9, 131 (008). [1] A. Iranmanesh, O. hormali. Journal of Compuaional and Theoreical Nanoscience, (in press). 71

6 7 [13] A. Iranmanesh, B. Soleimani, A. Ahmadi. Journal of Compuaional and Theoreical Nanoscience, 4, 147 (007). [14] A. Iranmanesh, Y. Alizadeh and S. Mirzaei, Diges Journal of Nanomaerials and Biosrucures. 4, 7 ( 009). [15] A. Iranmanesh, B. Soleimani. MATCH Commun. Mah Compu. Chem. 57, 51 (007). [16] A. Iranmanesh, O. hormali. Journal of Compuaional and Theoreical Nanoscience. 5, 131 (008). [17] A. Iranmanesh, Y. Pakravesh. Ars Combinaoria. 7, 3606 (007). [18] A. Iranmanesh, B. Soleimani, MATCH Commun. Mah. Compu. Chem. 57, 51 (007). 7