Abstract
The connective eccentric index of a graph is a topological index involving degrees and eccentricities of vertices of the graph. In this paper, we have studied the connective eccentric index for double graph and double cover. Also we give the connective eccentric index for some graph operations such as joins, symmetric difference, disjunction, and splice of graphs.
1. 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 of by . For , the distance between and in is defined as the length of the shortest path between and in and is denoted by . For a given vertex of , the eccentricity is the largest distance from to any other vertices of . The sum of eccentricities of all the vertices of is denoted by [1]. If any vertex is adjacent to all the other vertices of then is called a well-connected vertex. Thus, if is a well-connected vertex, then . 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 are the orbits of under its natural action on and , . In particular, if is a regular graph, then , where is the number of vertices of , which is a -regular graph, and is the radius of . Let , , , , denote the complete graph with vertices, the cycle on vertices, -dimensional hypercube, -sided prism, and the -sided antiprism, respectively. It can be easily verified that the explicit formulae for the connective eccentric index of , , , , are as follows.
Proposition 1. Consider the following
Proposition 2. Consider the following
Proposition 3. Consider the following
Proposition 4. Consider the following
Proposition 5. Consider the following
Several studies on different topological indices related to graph operations of different kinds are available in the literature [7–11].
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].
2. Main Results
In this section, first we define and then compute eccentric connectivity index of double graph and double cover graph.
2.1. 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 , and the vertices of are given by the two sets and . Thus, for each vertex , there are two vertices and in . The double graph includes the initial edge set of each copy of , and, for any edge , two more edges and are added. The graph and its double graph is shown in Figure 1.
Theorem 6. The connective eccentric index of the double graph is given by , where is the number of vertices with degree , that is, of eccentricity one.
Proof. From the construction of double graph, it is clear that , where and are the corresponding clone vertices of . Also we can write when and when .
Thus the connective eccentric index of double graph is
Let be a simple connected graph with . The extended double cover of , denoted by , is the bipartite graph with bipartition where and in which and are adjacent if and only if either and 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 is illustrated in Figure 2.
Theorem 7. The connective eccentric index of the extended double cover satisfies the inequality
Proof. If is a graph with vertices and edges, then, from definition, the extended double cover graph consists of vertices and edges and and for .
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.
2.2. Join
The join of two graphs and , with disjoint vertex sets , and edge sets , , is the graph union together with all the edges joining and ; that is, consists of the vertex set and edge set .
Theorem 8. Let and be two graphs without well-connected vertices. Then, .
Proof. Let and be the numbers of vertices in and , respectively. Thus, . For vertices and , it holds that and . Since none of and contains well-connected vertices, then, for every , . So, from definition of connective eccentric index, we have from where the desired result follows.
Now we generalize the above result for disjoint graphs .
Theorem 9. Let be graphs with disjoint vertex sets and edge sets , , without well-connected vertices. Then,
Proof. From definition of join, we have and since none of contains any well-connected vertex, we have for all .
Thus the connective eccentric index of is given by
from where the desired result follows.
The following corollaries are direct consequences of the theorem.
Corollary 10. If denotes the join of copies of , then
Corollary 11. Let be the complete -partite graph [see Figure 3] having number of vertices. Here the vertex set can be partitioned into subsets such that and , , , . From definition of join it is clear that is the join of empty graphs with number of vertices. Then the connective eccentric index of is given by
2.3. Symmetric Difference
Let and be two graphs with vertex sets and and edge sets and . Then the symmetric difference of and , denoted by , is the graph with vertex set in which any two vertices are adjacent to whenever is adjacent to in or is adjacent to in , but not both. From definition of symmetric difference, the degree of a vertex of is given by [12]
Theorem 12. The connective eccentric index of the symmetric difference of two graphs and is given by where none of and 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, , for all vertices [7, 8].
Thus the connective eccentric index of symmetric difference of two graphs and is given by
from where the desired result follows.
2.4. Disjunction
The disjunction of two graphs and is the graph with vertex set in which is adjacent to whenever is adjacent to in or is adjacent to in . Obviously, the degree of a vertex of is given by [7, 8]
Theorem 13. The connective eccentric index of the disjunction of two graphs and is given by where none of and 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 of two graphs and is computed as from where the desired result follows.
2.5. Splice
Let and be two graphs . Let and be two given vertices of and , respectively. A splice of and at the vertices and is obtained by identifying the vertices and in the union of and and is denoted by [15]. Different topological indices of the splice graphs have already been computed [16]. Let ; then denotes the eccentricity of as a vertex of denotes the eccentricity of as a vertex of , and denotes the eccentricity of as a vertex of . Let be the degree of the vertex . Then the connective eccentric index of is computed as follows.
Theorem 14. The connective eccentric index of splice of and is given by
Proof. For any vertex , Sharafdini and Gutman [16] showed that Similarly, for any vertex , Thus, the desired result follows from the definition of connective eccentric index.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.