Bulletin of the Chemists and Technologists of Macedonia, Vol., No., pp. 9 (4 GHTMDD 444 ISSN 5 6 Received: February, 4 UDC: 547 : 54 Accepted: March 8, 4 Original scientific paper WIENER INDICES OF BENZENOID GRAPHS* Damir Vukičević, Nenad Trinajstić Department of Mathematics, University of Split, Teslina, HR- Split, Croatia The Rugjer Bošković Institute, P. O. Box 8, HR- Zagreb, Croatia trina@rudjer.irb.hr Formulae are derived for Wiener indices of a class of pericondensed benzenoid graphs consisting of three rows of hexagons of various lengths. A corresponding computer program is prepared. Results obtained are checked against numbers computed by Pascal-oriented pseudocode that is based on ring-matrices of benzenoid graphs. This program can compute the Wiener index of any benzenoid system and not just for pericondensed benzenoids considered in this paper. Keywords: benzenoid graph; pericondensed benzenoids; ring-matrix; Wiener index INTRODUCTION *Dedicated to the th anniversary of this journal. The Wiener index W is the first topological index (graph-theoretical invariant to be used in chemistry []. It was introduced in 947 by Harold Wiener, then a chemistry student at the Brooklyn College, as the path number for characterization of alkanes [,]. In a chemical language, the Wiener index is equal to the sum of all shortest carboncarbon bond paths in a molecule. In a graphtheoretical language, the Wiener index is equal to the count of all shortest distances in a (molecular graph. First mathematical definition of this index, based on the concept of graph-theoretical distance as encoded in the distance matrix [4] is due to Hosoya [5]. Since its inception the Wiener index was used in a numerous structure-property studies [6 ]. Explicit formulas for several classes of benzenoid graphs were also proposed in recent years [ 8]. Note that benzenoid graphs are graphtheoretical representions of benzenoid hydrocarbons [9, ]. In this paper we give formulas for the calculation of the Wiener index of pericondensed benzenoid graphs made up from three rows of hexagons of various lengths (in a chemical language a subclass of pericondensed benzenoids [9, ]. A pericondensed benzenoid graph is a benzenoid graph in which internal vertices appear, that is, vertices that belong to three hexagons. Coronene and a number of its derivatives, such as dibenzo[a,j]coronene, dibenzo [bc,kl]coronene, naphtho[,-a]coronene, enumerous derivatives of perylene, anthathrene, peropyrene, etc., such as benzo[ghi]perylene, dibenzo[fg,op]anthathrene, naphtho[,,8-bcd]peropyrene and many other kinds of benzenoids belong, for example, to the studied class of pericondensed benzenoids. They possess interesting mathematical, physical, chemical, biological and technological properties [ 5]. In the application section we give a Pascaloriented pseudocode by which the Wiener index for any benzenoid graph can be calculated.
4 D. Vukičević, N. Trinajstić Let abc,, N. Denote by (,, DERIVATION OF FORMULAS G a b c a pericondensed benzenoid graph given below: First, we need to prove several Lemmas that will imply our main result. Lemma : Let n N and let L( n be a benzenoid chain depicted below: c benzenoid rings b benzenoid rings a benzenoid rings n - times Its Wiener index is given by 6n 6n W( L( n = n. Proof: Label vertices of L( n as on the following benzenoid chain: y y... y y n Also denote x x x x n... {, } {, } n. {,,..., n } { } X = x x x Y = y, y,..., y. n W( Ln ( = duv (, duv (, duv (, uv X uv Y u X v Y n n n n n i n n = ( j i ( j i ( i j ( j i n i= j= i i= j= i i= j= i= j= i 6n 6n = Lemma : Let n N. 896n 64n WGn nn n ( (,, = 98 6. Bull. Chem. Technol. Macedonia,,, 9 (4
Wiener indices of benzenoid graphs 5 Proof: Label vertices of Gn (, nn, as on the following pericondensed benzenoid graph: x 4, x 4, x 4, x 4,n x, x x,, x, x, x, x,n x,n x, x, x,... x,n Also denote WGn ( (, nn, = { } { } {,,,..., n } {,,,..., n } {,,,..., n } { n } S = x x x x,,,, S = x x x x,,,, S = x x x x uv, S S uv, S S4 u S S v S S4,,,, S = x, x, x,..., x. 4 4, 4, 4, 4, = duv (, duv (, duv (, = WLn ( ( WLn ( ( duv (, duv (, duv (, u S u S u S v S4 v S v S n n ( ( (( ( = W( L( n i j 6( n 4 n i= j= n n n n ( i j ( n ( i j ( n 8. i= j= i= j= Using the Lemma and eliminating the signs of the absolute value, we get ( (, n, n W G n = ( ( ( n ( ( ( ( ( n 6 6 = ( n n i b b ( i j ( j i ( 6( n (( 4 ( n i= j= i= j= i n i b b ( i j ( j i 8( n i= j= i= j= i ( ( ( n i b b ( i j ( i= j= i= j= i j i ( n. Glas. hem. tehnol. Makedonija,,, 9 (4
6 D. Vukičević, N. Trinajstić After some involved algebraic manipulations, we get 896n 64n WGn nn n ( (,, = 98 6. Lemma : Let abc,, Nsuch that b< c a. we have a 6a 5b b c W( G( abc,, = 6 a 6ab 6b ab 64 6c 6ac 6a c 6bc 64abc b c c 6 ac. Proof: Label vertices of Gabc (,, as on the following pericondensed benzenoid graph: x 7,b x 8,b x 9,b x,b Also denote...... x 7, x 8, x 9, x, x 7, x 8, x 9, x, x, x x,4 x, x x,4 x,,, x, {,,, } {,,, } S = x x x x,,.,4 S = x x x x,,,,4 x, x = {,,..., ( c b } = {,,..., ( c b } = {,,..., ( a b } = {,,..., ( a b } S x x x,,, S x x x 4 4, 4, 4, S x x x 5 5, 5, 5, S x x x 6 6, 6, 6, {,,..., b} {,,..., b} {,,..., b} { b} S = x x x 7 7, 7, 7, S = x x x 8 8, 8, 8, S = x x x 9 9, 9, 9, S = x, x,..., x.,,,,... x,(c b - - x 4,(c - b - - x 4, x 4,... x 5, x 5,... x 5,(a b - - x 5, 6, x x 6, 6,... x 6,(a b - - Bull. Chem. Technol. Macedonia,,, 9 (4
( (,, W G a b c Wiener indices of benzenoid graphs 7 = d( u, v d( u, v d( u, v { u, v} S S S7 S8 S9 S { u, v} S7 S8 S S S4 { u, v} S7 S8 S uv, S9 S S S5 S6 uv, S9 S S u S S4 u S S4 v S5 S6 v S d( u, v d( u, v d( u, v u S v S5 S6 = d( uv, d( uv, d( uv, d( uv, { } { } u S S4 u S5 S6 v S7 S8 S9 S v S7 S8 S9 S ( (,, ( ( ( ( ( ( ( ( ( ( ( ( c b a b c b ( i= j= i= ( a b b c b ( i= i= j= ( = W G b b b W L a W L b W L c W L b ( 4i 4j 5 6 6 7 4 5 6 4 5 6 7 8i ( a b b i= j= ( i j 4 4 4 5 5 6. ( ( 4 5 6 4 5 6 7 8i 4i 4j 4 5 5 6 Using Lemmas and, we get ( (,, W G a b c 896b 64b 6a 6a 6b 6b 98 6b a b 6c 6c 6b 6b c b ( c b ( a b ( 4i 4j 5 6 6 7 ( 4 5 6 4 5 6 7 8i i= j= = ( c b ( a b ( a b ( ( i j i= ( c b b i= i= j= b i= j= ( 4 5 6 4 5 6 7 8i 4i 4j 4 5 5 6 4 4 4 5 5 6. After a straightforward algebra, we get a 6a 5b b c W( G( abc,, = 6 a 6ab 6b ab 64 6c 6ac 6a c 6bc 64abc b c c 6 ac. Glas. hem. tehnol. Makedonija,,, 9 (4
8 D. Vukičević, N. Trinajstić Lemma 4: Let abc,, Nsuch that c b< a. W ( G( a, b, c 98a 6a b b = 4a 4ab 8a b 8b 8ab 8c ac 8a c 8bc 4b c 6c 8ac 4bc. 4c Proof: Note that G( a, b, c is a subgraph of pericondensed benzenoid graph G( a, b, b Gabc and G( a, b, b as on the following graph:. Label vertices of graphs (,,... x 5, x 5,... x,c x x 5, 5,t x,c x,c x 4,c x, x, x, x, x, x, x, x 7,x x, x 4,... x 4, x, x 4, x 6, x 6,x 6, 7, x 7, x 7,t x 6,t x 8, x 8, x 8, x 8,t... x, x, x, x 9, x 9, x x 9, 9,a - b - x,a - b - where t = ( b c. Also denote {,,..., c} {,,..., c} {,,..., c} {,,..., c} S = x x x,,, S = x x x,,, S = x x x,,, S = x x x 4 4, 4, 4, = {,,..., ( b c } = {,,..., ( b c } = { x ( b c } = {,,..., ( b c } S x x x 5 5, 5, 5, S x x x S 6, 6, 6, 6, 6, 6, S7 x7, x7, x7, S8 = { x8,, x8,,..., x8,( b c } {,,..., ( a b } { ( a b } S = x x x 9 9, 9, 9, S = x, x,..., x.,,, Bull. Chem. Technol. Macedonia,,, 9 (4
Wiener indices of benzenoid graphs 9 ( (,, W G abc {, } = d( u, v d( u, v d( u, v = uv S S S S4 S5 S6, S6, S7 S8 S9 S u S5 u S5 v S S S S4 v S6, S6, d( u, v d( u, v d( u, v d( u, v d( u, v d( u, v (, {, } u S5 u S5 u S5 u v S5 u S6, u S6, v S7 v S8 v S9 S v S v S6, S d u v u S6, u S6, v S S7 S9 v S4 S8 S ( ( ( b c (, d u v c = W( G( a, b, b ( 4 4i 4j 4 i= j= ( ( b c b c b c b c ( ( ( ( i j b c i j 8 b c i= j= i= j= ( ( b c b c ( i j 6( b c ( 4 ( b c 6 i= j= ( ( ( ( b c a b a b ( 4 i j ( 5 6 6 7 4i i= j= i= b c i c b ( i j ( ( b c i ( i i= j= i= i= ( a b ( b b a b i= i= i= i= ( i ( i ( 4 i 5 ( 4 i Using the last Lemma and eliminating the signs of absolute value, we get ( (,, W G abc ( ( = ( b c c = W( G( a, b, b ( 4 4i 4j 4 i= j= ( ( ( b c i b c b c ( i j ( j i ( b c i= j= i= j= i ( b c i b c b c ( i j ( j i 8( b c i= j= i= j= i ( ( ( ( ( ( b c i b c b c ( i j ( j i 6( b c ( b c 6 i= j= i= j= i ( b c a b a b ( 4 i j ( 5 6 6 7 4i i= j= i= b c i c b ( i j ( ( b c i ( i i= j= i= i= ( a b ( a b b b ( i ( i ( i ( i 4 5 4. i= i= i= i=. Glas. hem. tehnol. Makedonija,,, 9 (4
D. Vukičević, N. Trinajstić After a rather involved algebra, we get 98a 6a b b 4c W( G( a, b, c = 4a 4ab 8a b 8b 8 ab 8c ac 8a c 8bc 4b c6c -8ac -4bc. Lemma 5: Let ab, Nsuch that b a. 46b 6b W( G( a b a a a a ab a b b ab,, = 8 6 4 6 8 6. Proof: Denote by S the set of vertices marked by black dots on the following pericondensed benzenoid graph and denote by S the set of remaining vertices: ( ( ( { } ( b a ( ( W G aba,, = d uv, d uv, d uv, { } uv, S u S uv, S u S a ( ( ( i= j= ( ( = W G a, a, a 8i 8 j W L b a. Using Lemmas and, we get ( b a i= j= ( a ( a 896 64 W( G( abc,, = 98 6( a ( b a a 6( b a 6 ( 8i 8j ( b a. After involved, but straightforward algebraic manipulations, we get 46b 6b W( G( a b a a a a ab a b b ab,, = 8 6 4 6 8 6. Lemma 6: Let abc,, Nsuch that c a b. Bull. Chem. Technol. Macedonia,,, 9 (4
Wiener indices of benzenoid graphs 97a a 46b 6b 49c W( G( abc,, = 5 4a ab 8ab 8b 8ab 8c ac 4a c bc 8b c 4c 4ac 8 bc. Proof: Denote by S the set of vertices marked by black dots on the following pericondensed benzenoid graph, denote by S the single vertex marked by black square, denote by S the set of vertices marked by black triangles and by S 4 the set of remaining vertices: ( (,, = (, (, (, (, (, (, W G a b c d u v d u v d u v d u v d u v d u v { } ( b a ( (,, ( 6 6 ( b a c i= j= u, v S S4 u S u S u, v S u S u S v S S4 v S v S4 v S a = W G a a c i j i= j= ( ( ( ( i ( ( a c a c i j W L b a a c 4 5 4 i= Using Lemmas and 4, we get ( (,, W G abc ( b a a i= j= = i= ( a ( 6i 6j ( ( a c i. 98a 6a 4a 4a( a 8a ( a 8( a 8a( a = ( a- 4c 8c ac 8a c 8bc 4b c6c -8ac -4( a- c ( ( b a c 6 b a 6 b a ( ( a c ( a c i j ( b a i= j= a c 4 5 ( 4 i ( ( a c i. i= i= After some algebraic manipulations, we get { } ( Glas. hem. tehnol. Makedonija,,, 9 (4
D. Vukičević, N. Trinajstić 97a a 46b 6b 49c W( G( abc,, = 5 4a ab 8ab 8b 8ab 8c ac 4a c bc 8b c 4c 4ac 8 bc. From Lemmas 6 the main result follows. Theorem : Let abc,, N. ( (,, W G a b c THE MAIN RESULT 98c 6c b 4c 4cb 8c b b 4a 8b 8 cb, a b< c 8a ac 8c a 8ba 4b a6a -8ca -4ba 97c c 46b 5 4c cb 8c b 6b 49a 8b 8cb ac, a< c b 8a 4ca ba 8b a 4a 4ca 8ba 46b 8a a 6a 4ab, a= c b 6b 6ab 8b 6ab = c 6c 5b b a 6 c 6cb 6b cb 64, b< a c 6a 6ac 6c a 6ba 64abc b a a 6ca a 6a 5b b c 6 a 6ab 6b ab 64, b< c< a 6c 6ac 6a c 6bc 64abc b c c 6ac 97a a 46b 6b 5 4a ab 8a b 8b 8ab, c< a b 49c 8c ac 4a c bc 8b c 4c 4ac 8bc 98a 6a b b 4a 4ab 8a b 8b 8 ab, c b< a 4c 8c ac 8a c 8bc 4b c6c -8ac -4bc Bull. Chem. Technol. Macedonia,,, 9 (4
Wiener indices of benzenoid graphs APPLICATIONS Computer program based on the theorem Here we give a program written in Mathematica that uses the theorem from above to calculate values of G(a,b,c. a 6a 5b b c g [ a_, b_, c_ ] = 6 a 6ab 6b ab 64 6c 6ac 6a c 6bc 64abc b c c 6ac 98a 6a b b 4c g [ a_, b_, c_ ] = 4a 4ab 8a b 8b 8 ab 8c ac 8a c 8bc 4b c6c -8ac -4bc 46b 6b gaa [ _, b_, c_ ] = 8a a 6a 4ab 6ab 8b 6ab 97a a 46b 6b 49c gba [ _, b_, c_ ] = 5 4a ab 8ab 8b 8ab 8c ac 4a c bc 8b c 4c 4ac 8bc g a b c Which [ _, _, _] = [ <= && <, [,, ], < && <=, [,, ], == && <=, [, ], < && <=, [,, ], < && <, [,, ], < && <=, [,, ], <= && <, [,, ] ] = Array g, { 5,5,5} a b b c g c b a a c c b g b c b a a c c b g a a b b a a c g c b a b c c a g abc c a a b g b a b c c b b a g a b c ar Calculations Using this program we calculated values for G( a b c for five values of a:,,, a, b, c 5. These values are given below Glas. hem. tehnol. Makedonija,,, 9 (4
4 D. Vukičević, N. Trinajstić a = b\ c 4 5 7 44 78 6 5 65 944 498 76 97 8 756 57 4 485 7 967 94 948 5 79 574 88 44 7 a = b\ c 4 5 44 654 68 68 58 65 85 4 746 59 8 8 54 9 4 7 96 4 597 4 5 574 89 4 65 4 a = b\ c 4 5 78 68 594 5 74 944 4 5 88 54 79 54 4 967 4 545 94 67 5 88 4 579 994 457 a = 4 b\ c 4 5 68 5 86 456 498 746 88 978 464 756 894 9 4 94 597 94 7 48 5 44 65 994 44 498 a = 5 b\ c 4 5 6 58 74 456 5986 76 59 464 56 57 9 54 9 58 4 948 4 67 48 494 5 7 4 457 498 555 Bull. Chem. Technol. Macedonia,,, 9 (4
Wiener indices of benzenoid graphs 5 Use of the ring-matrix Let us describe the notion of the ring-matrix of a benzenoid graph by the following example. For the following multiply-connected benzenoid graph, that is, a benzenoid graph with a hole [6]: and the corresponding ring-matrix is given by: The connection between the ring-matrix and a benzenoid graph is illustrated below: Here we give a program, written in a Pascal-based pseudocode, that calculates Wiener index of a benzenoid system from its ring-matrix. We have used this program to verify our formulas. The results given by this program coincide with the results given in subsection 4.. Here is the program: Glas. hem. tehnol. Makedonija,,, 9 (4
6 D. Vukičević, N. Trinajstić. Read dimensions of the ring-matrix and store ring-matrix in array a (locations a(,-a(na,ma.. Add columns a(x, and a(x,ma, <=x<=na consisting of zeros and add rows a(,y and a(na,y consisting of zeros.. Set all entries of array g to zero. 4. For each <=i<=na and <=j<=*ma do if ( ((i% == AND ((j% == {symbols a%b here and forward mean a mod b } if ( (a[i - ][j/] == AND (a[i][j/] == AND (a[i][j/ ] == {symbols a/b here and forward mean integer division} g[(i - * ( * ma j][(i - * ( * ma j] = ; else if ( a[i-][j/] if ( j > g[(i - * ( * ma j][(i - * ( * ma j - ] = ; if ( j < * ma g[(i - * ( * ma j][(i - * ( * ma j ] = ; if ( a[i][j/] if ( j > g[(i - * ( * ma j][(i - * ( * ma j - ] = ; if ( i < na g[(i - * ( * ma j][i * ( * ma j] = ; if ( a[i][j/ ] if ( j < * ma g[(i - * ( * ma j][(i - * ( * ma j ] = ; if ( i < na g[(i - * ( * ma j][i * ( * ma j] = ; if ( ((i% == AND ((j% == if ( (a[i][j/] == AND (a[i-][j/ - ] == AND (a[i-][j/] == g[(i - * ( * ma j][(i - * ( * ma j] = ; else if ( a[i][j/] Bull. Chem. Technol. Macedonia,,, 9 (4
Wiener indices of benzenoid graphs 7 if ( j < * ma g[(i - * ( * ma j][(i - * ( * ma j ] = ; if ( j > g[(i - * ( * ma j][(i - * ( * ma j - ] = ; if (a[i-][j/ - ] if ( j > g[(i - * ( * ma j][(i - * ( * ma j - ] = ; if ( i > g[(i - * ( * ma j][(i - * ( * ma j] = ; if (a[i-][j/] if ( j < * ma g[(i - * ( * ma j][(i - * ( * ma j ] = ; if ( i > g[(i - * ( * ma j][(i - * ( * ma j] = ; if ( ((i% == AND ((j% == if ( (a[i][j/] == AND (a[i-][j/] == AND (a[i-][j/ ] == g[(i - * ( * ma j][(i - * ( * ma j] = ; else if ( a[i][j/] if ( j < * ma g[(i - * ( * ma j][(i - * ( * ma j ] = ; if ( j > g[(i - * ( * ma j][(i - * ( * ma j - ] = ; if (a[i-][j/] if ( j > g[(i - * ( * ma j][(i - * ( * ma j - ] = ; if ( i > g[(i - * ( * ma j][(i - * ( * ma j] = ; if (a[i-][j/ ] if ( j < * ma Glas. hem. tehnol. Makedonija,,, 9 (4
8 D. Vukičević, N. Trinajstić g[(i - * ( * ma j][(i - * ( * ma j ] = ; if ( i > g[(i - * ( * ma j][(i - * ( * ma j] = ; if ( ((i% == AND ((j% == if ( (a[i - ][j/] == AND (a[i][j/ - ] == AND (a[i][j/] == g[(i - * ( * ma j][(i - * ( * ma j] = ; else if ( a[i-][j/] if ( j > g[(i - * ( * ma j][(i - * ( * ma j - ] = ; if ( j < * ma g[(i - * ( * ma j][(i - * ( * ma j ] = ; if ( a[i][j/ - ] if ( j > g[(i - * ( * ma j][(i - * ( * ma j - ] = ; if ( i < na g[(i - * ( * ma j][i * ( * ma j] = ; if ( a[i][j/] if ( j < * ma g[(i - * ( * ma j][(i - * ( * ma j ] = ; if ( i < na g[(i - * ( * ma j][i * ( * ma j] = ; 5. Eliminate all colums and rows from g that contain entry. 6. Dijkstra's algorithm [7] is used to calculate distances between vertices. 7. Distances between vertices are summed up. REFERENCES Bull. Chem. Technol. Macedonia,,, 9 (4
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