376 _Acta Chim. Slov. 2016, 63, 376-379_doi: I0.i7344/acsi.20i6.2378
Scientific paper
The Eccentricity Version of Atom-Bond Connectivity Index of Linear Polycene Parallelogram Benzenoid
ABC5(P(n,n))
Wei Gao,1 Mohammad Reza Farahani2 and Muhammad Kamran Jamil3 *
1 School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China.
2 Department of Applied Mathematics of Iran University of Science and Technology (IUST),
Narmak, Tehran 16844, Iran.
3 Department of Mathematics, Riphah Institute of Computing and Applied Sciences (RICAS), Riphah International University,
14 Ali Road, Lahore, Pakistan.
* Corresponding author: E-mail: m.kamran.sms@gmail.com Phone: +923464105447
Received: 23-02-2016
Abstract
Among topological descriptors, connectivity indices are very important and they have a prominent role in chemistry. The atom-bond connectivity index of a connected graph G is defined as ABC(G) = y J1' '' where dv denotes the degree of vertex v of G and the eccentric connectivity index of the molecular graph G is defined as £(G) = x e(v ), where efv) is the largest distance between v and any other vertex u of G. Also, the eccentric atombond connectivity index of a connected graph G is equal to ABC5(G) = v
In this present paper, we compute this new Eccentric Connectivity index for an infinite family of Linear Polycene Parallelogram Benzenoid.
Keywords: Molecular graph, Atom-bond connectivity index; Eccentricity connectivity index, Linear Polycene Parallelogram Benzenoid
1. Introduction
Let G = (V E) be a graph, where V(G) is a non-empty set of vertices and E(G) is a set of edges. In chemical graph theory, there are many molecular descriptors (or Topological Index) for a connected graph, that have very useful properties to study of chemical molecules.1-4 This theory had an important effect on the development of the chemical sciences.
A topological index of a graph is a number related to a graph which is invariant under graph automorphisms. Among topological descriptors, connectivity indices are very important and they have a prominent role in chemistry.
One of them is Atom-Bond Connectivity (ABC) index of a connected graph G = (V,E) and defined as
(1)
where dv denotes the degree of vertex v of G, that introduced by Furtula et.al.5'6
On the other hands, Sharma, Goswami and Madan7 (in 1997) introduced the eccentric connectivity index of the molecular graph G as
i(G)=Yß, xs(v),
(2)
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where e(u) is the largest distance between u and any other vertex v of G. If x,yeV(G), then the distance d(x,y) between x and y is defined as the length of any shortest path in G connecting x and y. In other words, is maximum distance with first-point v in G.
e(v) = Max{d(v, u) | VveV(G)}
(3)
The Eccentric Connectivity polynomial of a graph G, was defined by Alaeiyan, Mojarad and Asadpour as follows:8,9
ECP(G,x)= Yj
vef(G)
,ecc( V')
(4)
Alternatively, the eccentric connectivity index is the first derivative of ECP(G;x) evaluated at x = 1. Now, by combine these above topological indexes, we now define a new version of ABC index as:10
(5)
cal index for an infinite family of Linear Polycene Parallelogram Benzenoid.
2. Main Results and Discussion
In this section, we computed the Eccentric atombond connectivity index ABC5 of an infinite family of Linear Polycene Parallelogram of Benzenoid graph,19 by continuing the results from.8,9,18,19 This Molecular graph has 2n(n + 2) vertices and 3n2 + 4n -1 edges.
For further study and more detail representation of Linear Polycene Parallelogram of Benzenoid P(n,n), see.8,9,18,19 Also, reader can see the general case of this Benzenoid molecular graph in Figure 1.
The general representation of Linear Polycene Parallelogram of Benzenoid P(n,n) is shown in Figure1.
Theorem 1. Let P(n,n) (VneN) be the Linear Polycene Parallelogram of benzenoid. Then the Eccentric atom-bond Connectivity index ABC5 of P(n,n) is equal to:
ABCn(P(n.n))=4

I Sn-4i-l	I Sjj—4r"-3
^4r-2(4ii + ])i+4«(4« + l} +J4r-2(4n-l)i t4h(4h-
Sn -4/ -S	i_ I	«»^¥-7__
2(Kn-3)i	+ + : y4/!-2(Rn +!)/ +2(s«--lOil +3)
MJJL+fa-ij^El-2(k-n1 4n*'
(2)i+1)	n
Un
-KJI + 5
(6)
We denote this new index of a connected graph G (Eccentric atom-bond connectivity index) by ABC5(G) (Since it is fifth definition of ABC index). For more details about the Atom-Bond Connectivity and Eccentricity connectivity indices see paper series.11-18
The aim of this paper is to exhibit this new topologi-
Proof: Let (Vn>1) P(n,n) depicted in Figure 1 be the general representation of Linear Polycene Parallelogram Benzenoid graph with 2n(n+2) vertices, such that 4n + 2 of them have degree two and 2n2-2 have degree three (V(P(n,n) = V2UV3). Thus there are 3n2+4n-1 (=V2[2(4n + 2) + 3(4n2-2)]) edges.
Table 1. Eccentric connectivity index for all vertices of Linear Polycene Parallelogram Benzenoid graph P(n,n).8,9
2n+l	2n+l	2n+2		4n-5	4n-4	4n-3	4n-2	4n-l
2n	2n+l	2n+2		4n-5	4n-4	4n-3	4n-2	
2n	2n+l	2n+2 ...		4n-5	4)1-4			
2n	2n+l	2)i+2						
								
								
								
2n	2n+l	2)i+2						
	2n+I							
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Acta Chim. Slov. 2016, 63, 379-379	377
Figure 1. WneN the general representation of Linear Polycene Parallelogram of Benzenoid P(n,n) and the eccentric connectivity of its vertices.
Now by refer to,8'9'16 we have the maximum eccentric connectivity and minimum eccentric connectivity for a veV(P(n,n)) as Max£(vj = 4n-1 andMine(vj=2n.
Now by according to Figure 1 and Table 1, it is easy see that:
• For all vertices with degree two in P(n,n), the ec-
centricity are equal to 4n-1, 4n-2, 4n-4, 4n-6,..., 2n + 2, 2n + 1 . • For all other vertices with degree three P(n,n) (dv =
3), the eccentricity are equal to 4n-3 until 2n. Thus, we have following computations by using Figure 1 and results in Table 1 as:
ABCs(P(n,n))=
£ lfW+4')-2
y ßBH+ T pQH+ v
M1') -•£('("»11 4'MV) «i'st|f(-j||) 40E(V)
= I(4n ~2i )+(4pi -2/ t l)-2 +4™-i (4«-2/)+(4n-2t-l)-2 + (2u + f)+(2n +1)-2 (4n-2i )(4n -2/ tI)	(4M-2/)(4fi-2f-l]: y (2m + 1)(2n +1)
(4/t—2f —2){4«—2f—1)	(4a-2i-2)(4ji-2i-3)	(2*)(2n]
(4a —2i —2){4n—2i
-2
-2i -1J-2
^ir
=4y ^»-^h^-^+I)^^
(4n-2i)(4jj-2/+l) L"
l(4ii-2iX4«-2i-l)-2++ 8i/h
(4a-H|P)i-2i-l) (2m+1)
+4Y-'ti)	. n l(4»-2/)+(4n-2,--l)-2+ fa
(4m-2-2)(4m-2M)	T (4m-2i)(4«-2i-l) 1 ' n
l^y.-i ( 8II-4J -3 + 8vb ^4/!-2(4m+1)/+4m (4n + l)	+	(2m+1)
^-i-; if a; 2
(7)

&i-4J-5
4ii-2(8M-3)i+2(gn--6H+l}

8ii-4I -1
n-1)
-2{8« + l)t+:(8>r-t0«+3)	"
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Acta Chim. Slov. 2016, 63, 376-379
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Finally, VneN, the fifth ABC index of Linear Polycene Parallelogram Benzenoid P(n,n) is equal to:
(8)
Here, we complete the proof of Theorem 1. ■
3.	Conclusions
In this paper, we consider a family of Linear Polycene Parallelogram Benzenoid and compute the Eccentric atom-bond Connectivity index ABC5. The Eccentric atombond Connectivity index ABC5 was defined as
ABCS(G) = £ JEEE^Hp, such that e(v) (Max{d(v,
u)NVveV(G)}) is the largest distance between v and any other vertex u of G.
4.	References
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Povzetek
Med topološkimi deskriptorji so indeksi povezanosti izredno pomembni in imajo vidno vlogo v kemiji. Indeks atomske povezanosti grafa G je definiran kot ABC(G )= £ pli \ kjer je dv stopnja vozlišča (točke) v od G ter je ecentrični
h. . h. ■. ■. V
indeks povezanosti grafa G definiran kot ZfG)= xs{v)> kjer je £(v) najdaljša razdalja med v in katerim koli voz, j
liščem u od G. Poleg tega je ecentrični indeks atomske povezanosti povezanega grafa G enak ABC5(G) =
= y k") + *(> )-2 .
V tem članku smo izračunali novi ecentrični indeks povezanosti za neskončno družino linearnih policenskih paralelo-gramskih benzenoidov.
Gao et al.: The Eccentricity Version of Atom-Bond Connectivity ...