Abstract
Formulas for calculations of the eccentric connectivity index and Zagreb coindices of graphs under generalized hierarchical product are presented. As an application, explicit formulas for eccentric connectivity index and Zagreb coindices of some chemical graphs are obtained.
1. Introduction
All graphs considered here are undirected, simple, and connected. For two vertices and of a graph , the distance is equal to the length of a shortest path connecting and . Suppose that and are the set of vertices and edges of , respectively. For every vertex , the edge connecting and is denoted by and ( for short) denotes the degree of in . The diameter of , denoted by , is the maximum distance among all pairs of vertices in the graph.
The first and second Zagreb indices are defined as respectively [1]. The applications of these graph invariants and their mathematical properties are reviewed in two important survey articles [2, 3]. When a topological index or a new graph operation is introduced, then the following mathematical questions are usually raised.(1)What are the extremal properties of this new topological index?(2)Is it possible to find exact formulas for this topological index under old and new graph operations?We refer to [4–7] for such questions about Zagreb group indices.
The Zagreb indices can be viewed as the contributions of pairs of adjacent vertices to certain degree-weighted generalizations of Wiener polynomials. The first and second Zagreb coindices were first introduced by Došlić [8]. They are defined as follows: In [9], the authors computed exact formulas for these graph parameters under some graph operations.
Now we define a vertex version of Zagreb indices as follows: The graph invariants and are called the first and second vertex Zagreb indices of .
The eccentricity is the largest distance between and any other vertex of . The total connectivity index of a graph is defined as . Also, the eccentric connectivity index of is defined as [10]. We refer to [11] for a good survey on this topological index.
A graph with a specified vertex subset is denoted by . Suppose and are graphs and . The generalized hierarchical product, denoted by , is the graph with vertex set and two vertices and are adjacent if and only if and or and ; see Figure 1. This graph operation was introduced recently by Barrière et al. [12, 13] and found some applications in computer science. We encourage the reader to consult [14–16] for the mathematical properties of the hierarchical product of graphs. The Cartesian product, , of graphs and has the vertex set and is an edge of if and or and [17].
We denote by and the path and cycle with vertices, respectively. Our notations are standard and can be taken from the standard books on graph theory.
2. Main Results
We start by introducing some notations that will be kept throughout this section. Let be a graph and . Following Pattabiraman and Paulraja [18], an path through in is an path in containing some vertices (not necessarily distinct from and ). Let denote the length of a shortest path through in . Notice that if one of the vertices and belongs to , then . Furthermore, let ; then and can be defined as follows:
Lemma 1 (see [13]). Let and be graphs with . Then one has the following:(a)if , then the generalized hierarchical product is the Cartesian product of and ;(b); ;(c) is connected if and only if and are connected;(d), = + , if ; , if .
Theorem 2 (see [19]). Let and be two connected graphs and let be a nonempty subset of . Then
Put . Then , , and . Therefore,
Corollary 3 (see [19]). Let and be two connected graphs. Then
Example 4. Let be the graph of truncated cube. Then , where , as shown in Figure 1. It is not difficult to check that , and so, by Theorem 2, we have .
Example 5. Consider the linear phenylene including benzene rings; see Figure 2. The linear phenylene is the graph , where On the other hand, by tedious calculations, one can check that So, by Theorem 2, we obtain
Theorem 6. Let and be two connected graphs and let be a nonempty subset of . Then, (1).(2) + + + .
Proof. Let and be two connected graphs and let be a nonempty subset of . The part (1) of this theorem is clear. To prove the second part, we notice that by definitions of generalized hierarchical product and , we have This completes the proof.
Corollary 7 (see [15]). Let and be two connected graphs and let be a nonempty subset of . Then
Proof. It follows from definition of that . On the other hand, . Therefore, So, by replacing and with and , respectively, in the above equation, the assertion follows immediately.
Corollary 8. Let and be two connected graphs and let be a nonempty subset of . Then
Proof. The assertion follows from Theorem 6, Corollary 7, and the fact that = .
Theorem 9 (see [15]). Let and be two connected graphs and let be a nonempty subset of . Then where is the set of all adjacent vertices of .
By Theorems 6 and 9 and this fact that , we can write the following.
Corollary 10. Let and be two connected graphs and let be a nonempty subset of . Then
Example 11. The molecular graph of dimer fullerene is the graph , where (Figure 3). On the other hand, it is not difficult to check that , , , , , , and . Thus, by the above results, we obtain(1) and .(2) and .
Example 12. Consider the linear phenylene including benzene rings (Figure 2). The linear phenylene is the graph , where ; see Figure 2. On the other hand, it is not difficult to check that , , , , , , and . Hence, by previous result, we obtain the following:(1) and ;(2) and = .
Example 13. Let be the graph of truncated cube. Then , where , as shown in Figure 1. Therefore,(1) and .(2) and .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The research of the first and second authors was supported by a grant from Ferdowsi University of Mashhad; (no. MA92295RAH). The research of the third author is partially supported by the University of Kashan under Grant no. 159020/40.