On Some Bounds and Exact Formulae for Connective Eccentric Indices of Graphs under Some Graph Operations

Let G be a simple connected graph with vertex set V(G) and edge set E(G). Let n and m be the number of vertices and edges of G, respectively. We denote the degree of a vertex V of G by d G (V). For u, V ∈ V(G), the distance between u and V in G is defined as the length of the shortest path between u and V in G and is denoted by d G (u, V). For a given vertex V of G, the eccentricity ε G (V) is the largest distance from V to any other vertices ofG.The sumof eccentricities of all the vertices of G is denoted by θ(G) [1]. If any vertex V ∈ V(G) is adjacent to all the other vertices of G then V is called a well-connected vertex. Thus, if V ∈ V(G) is a well-connected vertex, then


Introduction
Let  be a simple connected graph with vertex set () and edge set ().Let  and  be the number of vertices and edges of , respectively.We denote the degree of a vertex V of  by   (V).For , V ∈ (), the distance between  and V in  is defined as the length of the shortest path between  and V in  and is denoted by   (, V).For a given vertex V of , the eccentricity   (V) is the largest distance from V to any other vertices of .The sum of eccentricities of all the vertices of  is denoted by () [1].If any vertex V ∈ () is adjacent to all the other vertices of  then V is called a well-connected vertex.Thus, if V ∈ () is a well-connected vertex, then   (V) = 1.For example, all the vertices of a complete graph are well connected.
Recently, a number of topological indices involving vertex degree and eccentricity were subject to a lot of mathematical as well as chemical studies.A topological index of this type, introduced by Gupta et al. [2], was named as the connective eccentric index and was defined as Ghorbani [3] gave some bounds of connective eccentricity index and also computed this index for two infinite classes of dendrimers.De [4] reported some bounds for this index in terms of some graph invariants such as maximum and minimum degree, radius, diameter, first Zagreb index, and first Zagreb eccentricity index.In [5], Ghorbani and Malekjani computed the eccentric connectivity index and the connective eccentric index of an infinite family of fullerenes.In [6], Yu and Feng also derived some upper or lower bounds for the connective eccentric index and investigated the maximal and the minimal values of connective eccentricity index among all -vertex graphs with fixed number of pendent vertices.
In [3], Ghorbani showed that, for a vertex transitive graph , the connective eccentric index is given by where  1 , Several studies on different topological indices related to graph operations of different kinds are available in the literature [7][8][9][10][11].
In this paper, first we calculate connective eccentric index of double graph and double cover and hence the explicit formulae for the connective eccentric indices of join, symmetric difference, disjunction, and splice of graphs are obtained.For the definitions and different results on graph operations, such as join, symmetric difference, and disjunction, readers are referred to the book of Imrich and Klavžar [12].

Main Results
In this section, first we define and then compute eccentric connectivity index of double graph and double cover graph.

Connective Eccentric Index of Double Graph and Double
Cover.Let us denote the double graph of a graph  by  * , which is constructed from two copies of  in the following manner [13,14].Let the vertex set of  be () = {V 1 , V 2 , . . ., V  }, and the vertices of  * are given by the two sets  = { 1 ,  2 , . . .,   } and  = { 1 ,  2 , . . .,   }.Thus, for each vertex V  ∈ (), there are two vertices   and   in ( * ).The double graph  * includes the initial edge set of each copy of , and, for any edge V  V  ∈ (), two more edges     and     are added.The graph  3 and its double graph  * 3 is shown in Figure 1.

Theorem 6. The connective eccentric index of the double graph
* is given by   ( * ) = 4  () − 2( − 1)‖ − 1‖  , where ‖ − 1‖  is the number of vertices with degree  − 1, that is, of eccentricity one.Proof.From the construction of double graph, it is clear that , where V  ∈ () and   ,   ∈ ( * ) are the corresponding clone vertices of V  .Also we can write Thus the connective eccentric index of double graph  * is Let  = (, ) be a simple connected graph with  = {V 1 , V 2 , . . ., V  }.The extended double cover of , denoted by  * * , is the bipartite graph with bipartition (, ) where  = { 1 ,  2 , . . .,   } and  = { 1 ,  2 , . . .,   } in which   and   are adjacent if and only if either V  and V  are adjacent in  or  = .For example, the extended double cover of the complete graph is the complete bipartite graph.This construction of the extended double cover was introduced by Alon [13] in 1986.Extended double cover of  3 is illustrated in Figure 2.  Thus the connective eccentric index of extended double cover graph  * * is given by Now some exact formulae for the eccentric connectivity index of joins, symmetric difference, disjunction, and splice graphs are presented.
Then the symmetric difference of  1 and  2 , denoted by Theorem 12.The connective eccentric index of the symmetric difference  1 ⊕  2 of two graphs  1 and  2 is given by where none of  1 and  2 contains well-connected vertices.
Proof.Since the distance between any two vertices of a symmetric difference cannot exceed two, if none of the components contains well-connected vertices, the eccentricity of all vertices is constant and is equal to two; that is,   1 ⊕ 2 ( 1 ,  2 ) = 2, for all vertices ( 1 ,  2 ) [7,8].Thus the connective eccentric index of symmetric difference  1 ⊕  2 of two graphs  1 and  2 is given by from where the desired result follows.

Disjunction.
The disjunction  1 ∨  2 of two graphs  1 and  2 is the graph with vertex set ( 1 ) × ( 2 ) in which Obviously, the degree of a vertex ( 1 ,  2 ) of  1 ∨  2 is given by [7,8] where none of  1 and  2 contains well-connected vertices.
Proof.Since the distance between any two vertices of a disjunction cannot exceed two, if none of the components contains well-connected vertices, the eccentricity of all vertices is constant and equal to two [7,8].Then the connective eccentric index of the disjunction  1 ∨  2 of two graphs  1 and  2 is computed as Proof.For any vertex  ∈  1 , Sharafdini and Gutman [16] showed that Similarly, for any vertex  ∈  2 , Thus, the desired result follows from the definition of connective eccentric index.
International Journal of Combinatorics the explicit formulae for the connective eccentric index of   ,   ,   , Π  ,   are as follows.