A counting polynomial, called Omega Ω(G,x), was proposed by Diudea. It is defined on the ground of “opposite edge strips” ops. Theta Θ(G,x) and Pi Π(G,x) polynomials can also be calculated by ops counting. In this paper we compute these counting polynomials for a family of Benzenoid graphs that called Hexagonal trapezoid system Tb,a.

1. International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.5, October 2013
THETA Θ(G,X) AND PI Π(G,X)
POLYNOMIALS OF HEXAGONAL
TRAPEZOID SYSTEM TB,A
Mohammad Reza Farahani1
1
Department of Applied Mathematics,
Iran University of Science and Technology (IUST) Narmak, Tehran, Iran
ABSTRACT
A counting polynomial, called Omega Ω(G,x), was proposed by Diudea. It is defined on the ground of
“opposite edge strips” ops. Theta Θ(G,x) and Pi Π(G,x) polynomials can also be calculated by ops
counting. In this paper we compute these counting polynomials for a family of Benzenoid graphs that called
Hexagonal trapezoid system Tb,a.
KEYWORDS
Molecular graph, Benzenoid, Hexagonal trapezoid system, Omega Ω(G,x) polynomial, Theta Θ(G,x)
Polynomial, and Pi Π(G,x) polynomial.
1. INTRODUCTION
Mathematical calculations are absolutely necessary to explore important concepts in chemistry.
Mathematical chemistry is a branch of theoretical chemistry for discussion and prediction of the
molecular structure using mathematical methods without necessarily referring to quantum
mechanics. In chemical graph theory and in mathematical chemistry, a molecular graph or
chemical graph is a representation of the structural formula of a chemical compound in terms of
graph theory.
A topological index is a numerical value associated with chemical constitution purporting for
correlation of chemical structure properties, chemical reactivity or biological activity. Let G(V,E)
be a connected molecular graph without multiple edges and loops, with the vertex set V(G) and
edge set E(G), and vertices/atoms x,y  V(G). Two edges/bonds e=uv and f=xy of G are called codistant “e co f”, if and only if they obey the following relation: [1, 2]
d(v,x)=d(v,y)+1=d(u,x)+1=d(u,y)
Relation co is reflexive, that is, e co e holds for any edge e of G; it is also symmetric, if e co f
then f co e and in general, relation co is not transitive. If “co” is also transitive, thus an
equivalence relation, then G is called a co-graph and the set of edges is C(e):={f  E(G)| e co f}),
called an orthogonal cut (denoted by oc) of G. In other words, E(G) being the union of disjoint
orthogonal cuts for i≠j & i, j=1, 2, …, k, E G   C  C2  ...  Ck 1  Ck and Ci  Cj  . Klavžar
1
[3] has shown that relation co is a theta Djoković [4], and Winkler [5] relation. Let m(G,c) be the
DOI:10.5121/ijcsa.2013.3501
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2. International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.5, October 2013
number of qoc strips of length c (i.e. the number of cut-off edges) in the graph G. Omega
Polynomial Ω(G,x), [6-9] for counting qoc strips in G was defined by Diudea as
 G , x    m G , c  xc
c
The summation runs up to the maximum length of qoc strips in G. The first derivative (in x=1)
equals the number of edges in the graph
  G,1  c m  G, c   c
Other counting polynomials “Θ(G,x)” and “Π(G,x)“ was defined as
(G, x)   m(G, c)c.x c
c
(G, x)   m(G, c)c.x
E ( G ) c
c
where m(G,c) is the number of strips of length c. Ω(G,x) and Θ(G,x) polynomials count
“equidistant edges” in G and Π(G,x) “non-equidistant edges”. The first derivative (computed at
x=1) of these counting polynomials provide interesting topological indices:
 '(G,1)  c m(G, c)  c 2
 '(G,1)  c m(G, c)  c( E (G)  c)
From above equations, one can see that the first derivative of omega polynomial (in x=1), equals
the number of edges in the graph G, Ω’(G,1)=|E(G)|. And also, the first derivative of Pi
polynomial (in x=1), is equals |E(G)|2-Θ(G,1). In other words, from definition of above counting
polynomials (Omega Ω(G,x), Theta Θ(G,x) and Pi Π(G,x)), one can obtain
 '(G,1)2   '(G,1)   '(G,1)
The aim of this study is to compute Theta Θ(G,x) and Pi Π(G,x) polynomials of Hexagonal
trapezoid system graphs. Tb,a. Here our notations are standard and mainly taken from [6-26].
2. MAIN RESULTS
In this section by using Cut Method and Orthogonal Cut Method we compute Theta Θ(G,x) and Pi
Π(G,x) polynomials for a family of benzenoid graphs that called Hexagonal trapezoid system Tb,a.
Reader can see general representation of this family in Figure 1 and Reference [27]. A hexagonal
trapezoid system Tb,a (a  b  ) is a hexagonal system consisting a−b+1 rows of benzenoid chain
in which every row has exactly one hexagon less than the immediate row.
Theorem 2.1. Consider the hexagonal trapezoid system Tb,a (a  b  ) , Then Theta polynomial of
Tb,a is:
Θ(Tb,a,x)=4x2+6x3+…+2(a-b+1)xa-b+1+(b+1)xb+1+(b+2)xb+2+…+(a-1)xa-1 +axa+(a+1)xa+1 +(6b2b2)xa-b+2
And Theta index of Tb,a is
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3. International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.5, October 2013
Θ(Tb,a)=a3+b3-2ab2+ 9 2 a2+ 11 2 b2+2ab + 13 2 a- 9 2 b+5
Figure 1: A general representation of the hexagonal trapezoid system Tb,a (a  b  )
In general case the number of vertices/atoms and edges/bonds of the hexagonal trapezoid system
2 a 1
3a 1
3
9
b
Tb,a are equal to 2a+1  i 2b1 i =a2−b2+4a+2 and 2a  i 3b1 i = (a 2  b2 )  a   1 for all
2
2
2
a  b .
From Figures 1 and By using the Cut Method, one can see that there are two types of edges-cut of
hexagonal trapezoid system (we denote the corresponding edges-cut by Ci and Ci) and we have
following table.
Table 2.1. All quasi-orthogonal cuts of hexagonal trapezoid system Tb,a a  b  .
quasi-orthogonal cuts
The length of qoc strips
The number of qoc strips
Ci  i=1,…, a-b
i+1
2
Ca b 1
ci  i=1,…,a-b+1
a-b+2
a-i+2
2b
1
Thus, from Table 1, we compute Theta polynomial of hexagonal trapezoid system Tb,a as
 Tb,a , x   c m Tb,a , c  .c.x c
 C m Tb,a , Ci  .Ci x Ci  m Tb,a , Ca b1  .Ca b1x Cab1  c m Tb,a , ci  .ci .x ci
i
a b
i
  m Tb,a , Ci  .Ci x Ci  m Tb,a , Ca b 1  .Ca b1x Cab1 
i 1
a b 1
 m T
i 1
a b
a b 1
i 1
b ,a
, ci  .ci .x ci
i 1
 2  i  1 x i 1   2ba  4b  2b2  x a b 2 
 a  2  i x
a  2 i
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4. International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.5, October 2013
=4x2+6x3+…+2(a-b+1)xa-b+1+(b+1)xb+1+(b+2)xb+2+…+(a-1)xa-1
+axa+(a+1)xa+1 +(2b+4b-2b2)xa-b+2
  a  1 x a 1 
a b 1
 a  2  ix
i 2
i
 x a  2i    2ba  4b  2b 2  x a b  2
Also, Theta index is equal to
a b 1
 a b 1

   2ix i    a  2  i  x a  2i   2ba  4b  2b 2  x a b  2 
i 1
|
(Tb,a , x) |x1   i 2
x 1
x
a b 1
a b 1
i 2
i 1
 2  i2 
a b
 a  2  i
a b
2
 2b  a  b  2 
2
 2  i  1    a  1  i    a  1  2b  a  b  2 
i 1
2
2
2
2
i 1
a b
   3i 2  i  2a  2   a 2  2a  3   a  1  2b  a 2  b2  4  2ab  4a  4b 
2
i 1
2

3 a  b  a  b 
3
2
   a  b 

   a  1  a  b    a  b    a  b   a 2  2a  3

2
2 




+(a2+2a+1)+(2b3+2a2b-4ab2+8ab-8b2+4)
=(a3-b3-3a2b+3ab2+ 3 2 a2-3ab+ 3 2 b2 + 1 2 a- 1 2 b)-(a3-2a2b+ab2-a2-b2+2ab+a2
-ab-a+b)+(a3-a2b +2a2-2ab +3a-3b) +(a2+2a+1)
+(2b3+2a2b-4ab2+8ab-8b2+4)
=(a3-b3-2a2b+2ab2+ 7 2 a2+ 5 2 b2-6ab+ 9 2 a- 9 2 b)+(a2+2a+1)+(2b3+2a2b
-4ab2+8ab-8b2+4)
=a3+b3-2ab2+ 9 2 a2+ 11 2 b2+2ab + 13 2 a- 9 2 b+5
Theorem 2.2. Pi polynomial Π(Tb,a,x) and Pi index Π(Tb,a) are equal to
Π(Tb,a,x)=4xe-2+6xe-3+…+2(a-b+1)xe+b-a-1+(b+1)xe-b-1+(b+2)xe-b-2+…+(a-1)xe-a+1-1
+axe-a+(a+1)xe-a-1+(6b-2b2)xe+b-a-2 and
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5. International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.5, October 2013
Π(Tb,a)= 9 4 a4+ 9 4 b4+ 9 2 a2b2 + 25 2 a3- 5 2 b3 - 23 2 ab2+ 3 2 a2b+ 69 4 a2- 15 4 b2+ 5 2 ab+ 5 2 a+ 11 2 b-4
From Table 2.1 and analogous to Theta polynomial, we have
 Tb,a , x    c m Tb,a , c  .c.x
E (Tb ,a ) c
 c m Tb,a , ci  .ci .x eci  C m Tb,a , Ci  .Ci x eCi  m Tb,a , Ca b1  .Ca b1x eCab1
i

a b 1
 m T
ci ,i 1

i
a b
e ci
  m Tb,a , Ci  .Ci x eCi  m Tb,a , Ca b 1  .Ca b 1x eCab1
b , a , ci  .ci .x
Ci ,i 1
a b 1
a b
  a  2  i  x eia2  2 i  1 x ei1   2ba  4b  2b2  xeba2
i 1
i 1
=4x +6x +…+2(a-b+1)x
e-2
e-3
+(b+1)xe-b-1+(b+2)xe-b-2+…+(a-1)xe-a+1-1
e+b-a-1
+axe-a+(a+1)xe-a-1+(2b+4b-2b2)xe+b-a-2
Clearly, Pi index Π(Tb,a) can be derived from the definition of Pi polynomial Π(Tb,a,x) by
3
9
b
replacing the exponent e (=|E(Tb,a)|) by (a 2  b2 )  a   1 is equal to
2
2
2
2
3
9
b
Π(Tb,a)= (Tb,a , x) |x1 =  (a 2  b2 )  a   1   Tb,a 


2
2 
2
2
= 9 4 a4+ 9 4 b4+ 9 2 a2b2+ 81 4 a2+ b + 27 2 a3- 3 2 b3- 27 2 ab2+ 3 2 a2b+ 3 2 a2+ 3 2 b2
4
+ 9 2 ab+9a+b+1 –(a +b3-2ab2+ 9 2 a2+ 11 2 b2+2ab + 13 2 a- 9 2 b+5)
3
= 9 4 a4+ 9 4 b4+ 9 2 a2b2 + 25 2 a3- 5 2 b3 - 23 2 ab2+ 3 2 a2b + 69 4 a2- 15 4 b2+ 5 2 ab+ 5 2 a+ 11 2 b-4
An especial case of this family is triangular benzenoid Gn, that is equivalent with a hexagonal
trapezoid system T1,n. It is easy to see that the triangular benzenoid Gn n  has n2+4n+1
vertices and 3 2 n(n+3) edges (see Figure 2).
Theorem 2.3. Let Gn= T1,n be the triangular benzenoid. Then

Theta polynomial of Gn is equal to Θ(Gn,x)= Θ(T1,n,x)=6x2+9x3+…+3nxn +(n+5)xn+1

Theta index of Gn is equal to Θ(Gn)= Θ(T1,n)=n3+ 9 2 n2+ 13 2 n +7

Pi polynomial of Gn is equal Π(Gn,x)=6xe-2+9xe-3+…+3nxe-n+(n+5)xe-n-1
where e= 3 2 n(n+3)

Pi index of Gn is Π(Gn)= 9 4 n4+ 25 2 n3+ 93 4 n2- 13 2 n- 7 2
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6. International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.5, October 2013
Figure 2: Graph of triangular benzenoid G7 or T1,7.
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