Mohammad A Iranmanesh

Roslan Hasni

Saeid Alikhani

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Digest Journal of  Nanomaterials and Biostructures                     Vol. 6, No 2, April - June 2011, p. 699 - 702 
 
 
 
 
ON THE ECCENTRIC CONNECTIVITY INDEX OF CERTAIN  
MOLECULAR GRAPHS 
 
 
SAEID ALIKHANI*
G
, ROSLAN HASNIa, MOHAMMAD A. IRANMANESH 
Department of Mathematics,  Yazd University ,  89195-741, Yazd, Iran 
aSchool of Mathematical Sciences,  Universiti Sains Malaysia,11800 
Penang, Malaysia 
    
The eccentric connectivity index of a molecular graph  is defined as 
)()()(=)( veccvdegGVuG ∑ ∈ζ , where )(vecc  is defined as the length of a maximal 
path connecting v  to another vertex of G . In this paper we obtain some formulas for 
computing of this index for some specific graphs. 
  
(Received January 27, 2011; Accepted April 4, 2011) 
 
Keywords:  Eccentric connectivity index, Eccentric connectivity polynomial, graph  
     
 
1.  Introduction 
  
A simple graph ),(= EVG  is a finite nonempty set )(GV  of objects called vertices 
together with a (possibly empty) set )(GE  of unordered pairs of distinct vertices of G  called 
edges. In chemical graphs, the vertices of the graph correspond to the atoms of the molecule, and 
the edges represent the chemical bonds. 
   If )(, GVyx ∈  then the  distance ),( yxd  between x  and y  is defined as the length of 
a minimum path connecting x  and y . The  eccentric connectivity index of the molecular graph 
G , )(Gcξ , was proposed by Sharma, Goswami and Madan 8. It is defined as 
)()()(=)( ueccuGdegGVuG
c ∑ ∈ξ , where )(xGdeg  denotes the degree of the vertex x  in G  
and )}(|),({=)( GVxuxdMaxuecc ∈ , for details see 653 ,, . The radius and diameter of G  are 
defined as the minimum and maximum eccentricity among vertices of G , respectively. The 
eccentric connectivity polynomial of a graph G  is )()()(=),(
veccxvdegGVvxGECP ∑ ∈ , 
(see 21,  ). Therefore the eccentric connectivity index is the first derivative of ),( xGECP  evaluated 
at 1=x  (see 7). 
  The  corona of two graphs 1G  and 2G , as defined by Frucht and Harary in 
4, is the 
graph 21= GGG   formed from one copy of 1G  and |)1(| GV  copies of 2G , where the ith vertex 
of 1G  is adjacent to every vertex in the ith copy of 2G . The corona 1KG  , in particular, is the 
graph constructed from a copy of G , where for each vertex )(GVv∈ , a new vertex v′  and a 
pendant edge vv ′  are added. The  join of two graphs 1G  and 2G , denoted by 21 GG ∨  is a graph 
with vertex set )2()1( GVGV ∪  and edge set )1(|{)2()1( GVuuvGEGE ∈∪∪  and )}2(GVv∈ . 
                                                     
*Corresponding author: alikhani@yazduni.ac.ir 
700 
 
A graph is an  empty graph if contain no edges. The empty graph of order n  is denoted by nO . 
    In Section 2 we compute the eccentric connectivity polynomial for specific graphs 
denoted by )(mG  and 2)(1 GmG . In Section 3 we compute the eccentric connectivity polynomial 
for corona and join of two graphs. Using our results, we study the eccentric connectivity index of 
these kind of graphs. 
 
 
2.  Eccentric connectivity index of some specific graphs 
 
In this section we consider two certain graphs with specific construction and compute their 
eccentric connectivity indexes. 
Let 1+mP  be a path with vertices labeled by myyy ,,1,0  , for 0≥m  and let G  be any 
graph. Denote by )(0 mvG  (or simply )(mG , if there is no likelihood of confusion) a graph 
obtained from G  by identifying the vertex 0v  of G  with an end vertex 0y  of 1+mP  (see Figure 
0). For example, if G  is a path 2P , then )(2=)( mPmG  is the path 2+mP . Also, we denote the 
graph obtained from graphs 1G  and 2G  by adding a path mP  from a vertex in 1G  to a vertex of 
2G , by 2)(1 GmG . (Fig. 1). 
 
             
Fig. 1.  Graphs )(mG  and 2)(1 GmG , respectively. 
  
 Theorem 1 .  The eccentric connectivity index of )(0 mvG  is  
 
 )}.(1,)0,({)(
)(
=))(( ueccmvudMaxudeg
GVu
mG ++
∈
∑ζ  
  
    Proof. It is easy to see that the eccentricity of a vertex u  in a graph )(0 mvG  is 
)}(1,)0,({ ueccmvudMax ++ . Therefore we have the result by definition of eccentricity index. ■  
  
  We state and prove the following theorem which is about the eccentric connectivity index of 
2)(1 GmG : 
  
Theorem 2 .  The eccentric connectivity polynomials of 2)(1 GmG  is 
)}(21,)(1),({)2(
)}(11,)(2),({)()1(
=)2)(1(
ueccmaeccbuddeguMaxGVu
ueccmbeccaudMaxudegGVuGmG
+++∈
++++∈
∑
∑ξ
, 
 where ba,  are two vertices of 2)(1 GmG  in Figure 1.   
    Proof. We observe that the eccentricity of a vertex u  in a graph 1G  is 
701 
 
)}(11,)(2),({ ueccmbeccaudMax +++ . Also the eccentricity of a vertex u  in a graph 2G  is 
)}(21,)(1),({ ueccmaeccbudMax +++ . So we have the result by definition of eccentricity index.  
■  
3.  The eccentricity index of corona and join of two graphs 
 
In this section we consider the corona and the join of two graphs, and compute their 
eccentricity connectivity polynomials. Using our results, we obtain formulas for eccentricity index 
of corona and join of two graphs. 
  
Theorem 3 .  Let 1G  and 2G  be two graphs of order 1n  and 2n , respectively. Then  
 ])(
1
)2[()),2(),1((=),21(
veccx
Gv
xnxxGECPxGECPxxGGECP ∑
∈
+++ .  
   Proof. In the corona of two graphs 1G  and 2G , the degree of each vertices of 1G  increase by 
2n  and the eccentricity of these vertices increase by one. Also the degree of each vertices of 2G  
increase by one and the eccentricity of these vertices increase by two. Therefore we have 
 
 )().(
)21(
=)21(
veccxvdeg
GGVv
GGECP ∑
∈ 
  
 2)(1))((1)()2)((=
+++++ ∑∑ veccxvdegveccxnvdeg  
 ].)(
1
)2[()),2(),1((=
veccx
Gv
xnxxGECPxGECPx ∑
∈
+++ ■ 
 
  Let us denote graph 1KH   simply by 
*H . We have the following corollary: 
  
Corollary 1 .  Suppose that H  is a graph of order n . Then  
 ].)()[(1)),((=),*( veccx
Hv
xxxHECPxxHECP ∑
∈
++   
    Proof. It suffices to put 1=2 KG  in Theorem 3. Since 0=),1( xKECP , we have the result. ■  
  
  The following theorem give us the formula for )21( GGECP ∨ : 
  
Theorem 4 .  Suppose that 1G  and 2G  are two graphs with orders m  and n , respectively. Then 
we have  
 )).(
)2(
)(
)1(
(22=)21( wdeg
GVw
vdeg
GVv
mnxGGECP ∑∑
∈
+
∈
+∨  
  
    Proof. We observe that when we construct 21 GG ∨ , the degree of each vertices in 1G  increase 
by n  and the degree of each vertices in 2G  increase by m . Also observe that the eccentricity for 
every vertex is two. Therefore by definition of eccentric connectivity polynomial we have: 
 
 )().(
)21(
=)21(
veccxvdeg
GGVv
GGECP ∑
∨∈
∨  
702 
 
 2))((
2
2))((
1
= xmwdeg
Gw
xnvdeg
Gv
+
∈
++
∈
∑∑  
 )).(
)2(
)(
)1(
(22= wdeg
GVw
vdeg
GVv
mnx ∑∑
∈
+
∈
+ ■ 
As an example, by applying Theorem 4 for nOG =1  and mOG =2 , we have the following 
corollary for complete bipartite graphs (see 1 ). 
  
Corollary 2 . If 2, ≥nm , then the eccentricity connectivity polynomial of nmK ,  is 
22=),,( mnxxnmKECP .   
 
  Since the eccentric connectivity index is the first derivative of ),( xGECP  evaluated at 1=x , 
then we have the following theorem. 
  
Theorem 5 .  Let 1G  and 2G  be two graphs of orders m  and n , respectively. Then  
 ),(
)1(
1)()212(=)21( vecc
GVv
nmnmeeGGc ∑
∈
+++++ζ  
where 1e  and 2e  are the number of edges of 1G  and 2G , respectively.   
 
  
Theorem 6 .  Let 1G  and 2G  be two graphs of orders m  and n , respectively. Then we have  
 )).(
)2(
)(
)1(
2(2=)21( wdeg
GVw
vdeg
GVv
mnGGc ∑∑
∈
+
∈
+∨ζ  
        
 
References 
 
[1] S. Alikhani, M.A. Iranmanesh,  Digest Journal of nanomaterials and biostructures,  
      6(1), 253 (2011). 
[2] A.R. Ashrafi, M. Ghorbani and M. Hemmasi,  Digest Journal of Nanomaterials and  
       Biostructures 4(3),  483 (2009). 
[3] Shubo Chen, Weijun Liu, and Fangli Xia,  J. Comput. Theor. Nanosci. 8, 17 (2011). 
[4] R. Frucht and F. Harary,  Aequationes Math. 4, 322-324 (1970). 
[5] I. Gutman, O.E. Polansky,  Mathematical Concepts in Organic Chemistry, Springer-Verlag,  
       New York, (1986). 
[6] M. A. Johnson, G.M. Maggiora,  Concepts and Applications of Molecular Similarity, Wiley  
      Interscience, New York, (1990). 
[7] M.J. Morgan, et al., Discrete Mathematics (2010), doi:10.1016/j.disc.2009.12.013. 
[8] V. Sharma, R. Goswami, A. K. Madan, J. Chem. Inf. Comput. Sci. 37, 273 (1997). 
 
 
 


ON THE ECCENTRIC CONNECTIVITY INDEX OF CERTAIN MOLECULAR GRAPHS

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ON THE ECCENTRIC CONNECTIVITY INDEX OF CERTAIN MOLECULAR GRAPHS

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