Modified Eccentric Connectivity of Generalized Thorn Graphs

The thorn graph G of a given graph G is obtained by attaching t(> 0) pendent vertices to each vertex of G. The pendent edges, called thorns of G, can be treated as P 2 or K 2 , so that a thorn graph is generalized by replacing P 2 by P m and K 2 by K p and the respective generalizations are denoted by G P m and G K p . The modified eccentric connectivity index of a graph is defined as the sum of the products of eccentricity with the total degree of neighboring vertices, over all vertices of the graph in a hydrogen suppressed molecular structure. In this paper, we give the modified eccentric connectivity index and the concerned polynomial for the thorn graph G and the generalized thorn graphs G K p and G P m .


Introduction
Let  be a simple connected graph with vertex set () and edge set (), so that |()| =  and |()| = .Let the vertices of  be labeled as V 1 , V 2 , . . ., V  .For any vertex V  ∈ () the number of neighbors of V  is defined as the degree of the vertex V  and is denoted by   (V  ).Let (V  ) denote the set of vertices which are the neighbors of the vertex V  , so that |(V  )| =   (V  ).Also let   (V  ) = ∑ V  ∈(V  )   (V  ), that is, sum of degrees of the neighboring vertices of V  ∈ .The distance between the vertices V  and V  is equal to the length of the shortest path connecting V  and V  .Also for a given vertex V  ∈ (), the eccentricity   (V  ) is the largest distance from V  to any other vertices of  and the sum of eccentricities of all the vertices of  is denoted by () [1].The eccentric connectivity index of a graph  was proposed by Sharma et al. [2].A lot of results related to chemical and mathematical study on eccentric connectivity index have taken place in the literature [3][4][5].There are numerous modifications of eccentric connectivity index reported in the literature till date.These include edge versions of eccentric connectivity index [6], eccentric connectivity topochemical index [7], augmented eccentric connectivity index [8], superaugmented eccentric connectivity index [9], and connective eccentricity index [10].A modified version of eccentric connectivity index was proposed by Ashrafi and Ghorbani [11].
Similar to other topological polynomials, the corresponding polynomial, that is, the modified eccentric connectivity polynomial of a graph, is defined as so that the modified eccentric connectivity index is the first derivative of this polynomial for  = 1.Several studies on this modified eccentric connectivity index are also found in the literature.In [11], the modified eccentric connectivity polynomials for three infinite classes of fullerenes were computed.In [12], a numerical method for computing modified eccentric connectivity polynomial and modified eccentric connectivity index of one-pentagonal carbon nanocones was presented.In [13], some exact formulas for the modified eccentric connectivity polynomial of Cartesian product, symmetric difference, disjunction, and join of graphs were presented.Some upper and lower bounds for this modified eccentric connectivity index are recently obtained by the present authors [14]. 2 International Journal of Computational Mathematics The first and the second Zagreb indices of , denoted by  1 () and  2 (), respectively, are two of the oldest topological indices introduced in [15] by Gutman and Trinajstić and were defined as Let  = ( 1 ,  2 , . . .,   ) be a -tuple of nonnegative integers.The thorn graph   is the graph obtained from  by attaching   pendent vertices to the vertex V  ,  = 1, 2, . . .,  of .In this paper, we assume  1 =  2 = ⋅ ⋅ ⋅ =   = .These pendent vertices are termed as thorns.The concept of thorn graphs was first introduced by Gutman [16].A lot of studies on thorn graphs for different topological indices are made by several researchers in the recent past [17][18][19][20][21][22][23][24].Very recently, De [25,26] studied two eccentricity related topological indices, such as eccentric connectivity index and augmented eccentric connectivity indices, on thorn graphs.
The thorns of the thorn graph   can be treated as  2 or  2 , so that the thorn graph can be generalized by replacing  2 by   and  2 by   and the generalizations are, respectively, denoted by    and    .In the following section, we present the explicit expressions of the modified eccentric connectivity index of thorn graph   and its generalized forms    and    .

Evaluation of Modified Eccentric Connectivity Index
The eccentric connectivity index   () [2] and connective eccentric index   () [10] of a graph  are defined as The modified eccentric connectivity index   () [11] is defined as Total eccentricity index is defined as () = ∑ V∈()   (V).Total eccentricity index of the generalized hierarchical product of graphs has been studied by De et al. recently [14].
Since the modified eccentric connectivity index is likely to have an application in drug discovery process, therefore, we evaluate the index in comparison to some other well known indices in this section.The two graphs shown in Figure 1 have the same eccentricity connectivity index and connective eccentricity index, but they have different modified eccentric connectivity indices.
To evaluate the modified eccentric connectivity index (MECI) in terms of degeneracy and intercorrelation with other well known indices we compute different topological indices such as eccentric connectivity index (ECI), total eccentricity index (TEI), connective eccentricity index, (CEI) and augmented eccentric connectivity index (AECI) for octane isomers as given in Table 1.
Intercorrelation of modified eccentric connectivity index with some well known vertex eccentricity based topological indices is given in Table 2.

Main Results
First we find the modified eccentric connectivity index of the thorn graph   in terms of modified eccentric connectivity index of , total eccentricity of , and first Zagreb index of .
Here,   are the set of degree one vertices attached to the vertices V  in   and   ∩  = ,  ̸ = .Let the vertices of the set   be denoted by V 1 , V 2 , . . ., V  for  = 1, 2, . . ., .
Then the modified eccentric connectivity index of   is given by Now, Combining the above equations, we get   (  ) =   () + 2  () +  ( + 1)  () +  1 () The eccentric connectivity polynomial and total eccentricity polynomial of  are defined as   (, ) = ∑  =1   (V  )   (V  ) and (, ) = ∑  =1    (V  ) , respectively.It is easy to see that the eccentric connectivity index and the total eccentricity of a graph can be obtained from the corresponding polynomials by evaluating their first derivatives at  = 1.
Now we express the modified eccentric polynomial of a thorn graph in terms of eccentric connectivity polynomial, modified eccentric connectivity polynomial, and total eccentric polynomial of the parent graph.

International Journal of Computational Mathematics
Proof.Following the previous theorem, the modified eccentric connectivity polynomial of   is given by After addition, we get   (  , ) =   (, ) +  ( + 1)   (, ) It can be easily verified that expression (7) is obtained by differentiating (9) with respect to  and by putting  = 1.
Therefore the modified eccentric connectivity index of    is given by   ( Adding, we get Since for  = 2, the generalized thorn graph    reduces to the usual thorn graph   , Theorem 1 follows from Theorem 3 by substituting  = 2. In the following, we find the modified eccentric connectivity polynomial of the graph    in terms of eccentric connectivity polynomial, modified eccentric connectivity polynomial, and total eccentric polynomial of the parent graph .Theorem 4. For any simple connected graph , the   (   , ) is given by   (   , ) =   (, ) + ( − 1)( + 1)  (, ) + ( − 1) 2 (1 +  + ( − 2))(, ), where    is the graph obtained from  by attaching  complete graphs   at each vertex of .
Proof.From definition, the modified eccentric connectivity polynomial of    is given by Now, Similarly, Adding the above two, we get   (   , ) =   (, ) +  ( − 1) ( + 1)   (, ) It can also be verified that expression (11) is obtained by differentiating (15) with respect to  and by putting  = 1.Also Theorem 2 follows from Theorem 4 by substituting  = 2.

Figure 1 :
Figure 1: Two graphs with the same   and   , but with different   .

Table 1 :
Different topological indices of octane isomers.