Abstract
The eccentric connectivity polynomial (ECP) of a connected graph is described as , where and represent the eccentricity and the degree of the vertex , respectively. The eccentric connectivity index (ECI) can also be acquired from by taking its first derivatives at . The ECI has been widely used for analyzing both the boiling point and melting point for chemical compounds and medicinal drugs in QSPR/QSAR studies. As the extension of ECI, the ECP also performs a pivotal role in pharmaceutical science and chemical engineering. Graph products conveniently play an important role in many combinatorial applications, graph decompositions, pure mathematics, and applied mathematics. In this article, we work out the ECP of -sum of graphs. Moreover, we derive the explicit expressions of ECP for well-known graph products such as generalized hierarchical, cluster, and corona products of graphs. We also apply these outcomes to deduce the ECP of some classes of chemical graphs.
1. Introduction
Let be an -vertex simple and connected graph with the vertex set and the edge set . For a given graph , the order and size are symbolized by and , respectively. The degree of is the number of adjacent vertices to in , and it is represented by . For , the distance between and , denoted with , is defined as the length of the shortest path among and in , and the eccentricity is the largest distance among and any other vertex of . We use notions and for the -vertex path and cycle, respectively. The line graph denoted by of is the graph whose vertices are the edges of the original graph; two vertices and are connected if and only if they share a common end vertex in . The joint of graphs and is the graph union including all the edges joining and .
A molecular descriptor is a numeric measure of a graph which characterizes its topology. In organic chemistry, topological invariants have established many applications in pharmaceutical drug design, QSAR/QSPR studies, chemical documentation, and isomer discrimination. Some effective topological classes such as degree based, degree distance, eccentric connectivity indices, and so on are established as molecular invariants. In recent years, the study of eccentric invariants for chemical molecular structure has become one of the flourishing lines of research in theoretical chemistry.
The ECI of is a newly discovered distance-based topological invariant which was put forward by Sharma et al. [1] and is defined as follows:
In recent times, a modification of is used and famous as the total eccentricity index . It can be defined as
The study of polynomials has been a valuable tool to describe the complex and classical behavior of dynamical systems. Furthermore, the polynomial approaches have also been utilized to figure out the central problems such as robustness, stability, and controllability. Polynomial theory can also be implemented in nonlinear, uncertain, hybrid, and time-delay systems and model predictive control. In recent years, there have been numerous works on graph polynomials related with various topological indices. For the detailed discussion about the different types of polynomials such as characteristic, chromatic, edge cover, domination, matching, and clique polynomials and their related studies, we refer the readers to [2–7].
Ashraf and Jalali gave the concept of ECP in [8], which can be specified as
The total eccentricity polynomial of can be expressed as follows:
It is an easy exercise to see that the eccentric connectivity and total eccentricity indices can be acquired from their related polynomials by taking their first derivatives at .
Graph products play a significant role in many combinatorial applications, graph decompositions into isomorphic subgraphs, and not only in pure mathematics but also in applied mathematics. In [9, 10], Akhter and Imran investigated the degree-based topological invariants for -sum of graphs. De et al. [11] presented the results related to the total eccentricity index of some products of graphs. The ECI of -sum graphs in the form of different invariants has been computed in [12]. Došlić and Saheli [13] established the ECI of composite graphs. The ECP of several classes of composite graphs, including Cartesian product, symmetric difference, disjunction, join of graphs, and composition of graphs have been computed in [14]. For more information on different aspects of ECP, one can see [15–21].
Inspired by Yang et al. [22], we continue the research on finding the indices and polynomials of -sum of graphs. This article is arranged as follows. First, we find the ECP of the generalized hierarchical product of graphs, and as applications, we give explicit outcomes for chemical graphs such as a truncated cube and linear phenylene. Furthermore, we compute the ECP of -sum of graphs and give its applications in Section 3. Finally, we determine the ECP of the cluster and the corona products of graphs in the last section.
Proposition 1. (see [23]). Let and denote the -vertex cycle and path, respectively. Then,(1)(2)(3)(4)
2. Generalized Hierarchical Product
Barrière et al. [24] brought in the concept of the generalized hierarchical product, that is, a generalization of the (standard) hierarchical product and Cartesian product of graphs [25]. The generalized hierarchical product of and with is represented by . It has vertex set and if and or and . For , a path between and through is a -path in including some vertex (vertex could be the vertex or ). For , the distance between these vertices through , symbolized as , is the length of a smallest -path through . It is easily seen that, if one of the vertices and belongs to , then (see [26]). Now, in Lemma 1, we describe the distinct properties of this product.
Lemma 1. (see [24]). Let and be graphs with . Then,(a) and .(b)(c)(d).In the next theorem, we give the expression of the ECP of in terms of eccentric connectivity and total eccentricity polynomials of and .
Theorem 1. Let and be graphs with . Then,where .
Proof. From Lemma 1 and formula (3), we getThis finishes the proof.
The Cartesian product of graphs and has the vertex set and if and or and . By using Theorem 1, we can deduce the following theorem by putting .
Theorem 2. (see [14]). Let and be two connected graphs. Then,
Example 1. The molecular graph of truncated cube can be represented as a generalized hierarchical product of and (depicted in Figure 1), with . It is easy task to see that andThen by Theorem 1, we have .
Example 2. The linear phenylene with benzene ring is the graph that is also recognized by (depicted in Figure 2), with . By tedious computation, we get andThen by Theorem 1, we have
3. -Sum of Graphs
Eliasi and Taeri initiated the notion of -sum of a graph in [27], and they studied the Wiener index of it. Some explicit expressions of various PI indices have been presented for -sum graphs in [28]. Metsidik et al. [29] studied the Wiener indices of -sum graphs.
The subdivision graph of , symbolized by , can be sketched from by interchanging each edge of with a path . The line superposition graph of can be attained from by placing a new vertex into each edge of and then linking with edges of each pair of new vertices on adjacent edges of . The triangle parallel graph of is symbolized by , and it can be constructed from by interchanging each edge of with a triangle.
The total graph of is represented by , which has edges and vertices of as its own vertices, and adjacency in is described as the adjacency or incidence of the related elements (vertices and edges) of .
Let be one of the aforementioned operations , or . The -sum of graphs and is represented by . It has vertex set , and if and only if and or and .
Lemma 2 (see [12]). Let be a connected graph. If , then(a) and .(b)(c)
Theorem 3. Let and be graphs. Then,
Proof. Followed by formula (3) and Lemma 2, we haveCombining these results with (5), we get the desired result. This finishes the proof.
Lemma 3 (see [12]). Let be a connected graph. If , then(a) and .(b)(c)
Theorem 4. Let and be graphs. Then,
Proof. With the help of definition of the ECP and Lemma 3, we haveCombining these results with (5), we get following:This completes the proof.
Lemma 4. (see [12]). Let be a connected graph. If , then(a) and .(b)(c)
Theorem 5. Let and be graphs. Then,
Proof. By formula (3) and Lemma 4, we haveCombining these results with (5), we get the required result. This accomplishes the proof.
Lemma 5. (see [12]). Let be a connected graph. If , then(a) and .(b)(c)
Theorem 6. Let and be graphs. Then,
Proof. With the help of (3) and Lemma 5, we haveCombining these results with (5), we get the needed result. This finishes the proof.
Example 3. For , let be a zig-zag polyhex nanotube as shown in Figure 3. By definition, it is obvious that , where . By Theorem 3, we get
Example 4. Let be the hexagonal chain [30] with hexagonals (see Figure 4). From definition, it is obvious that , with . Then, by applying Theorem 3, we get
4. Cluster and Corona Product
The cluster product of and , symbolized by , is achieved by using copies of a rooted graph and a copy of and by specifying the root of the -th copy of with the -th vertex of , . The order and size of are and , respectively. If we take as the root vertex of and , then . Now with the above notions in hand and by applying Theorem 1, we get the following result.
Theorem 7. Let and be graphs such that . Then,
Proof. Since , and . So, using Theorem 1, we get the desired result.
Let be a star graph having vertices, with the root vertex of degree . For a given graph , the graph can be constructed by taking cluster product of with . This is famous as -fold bristled graph . With the help of Theorem 1, the ECP of -fold bristled graph can be evaluated.
Corollary 1. Let be a graph. Then,
Proof. Let , with the central vertex of degree in as the root vertex. Then, and for all . The desired expression is obtained with the help of above theorem.
The corona product of graphs and is a graph, which can be drawn by using copies of and a copy of and linking the -th vertex of to every vertex in -th copy of , . In Theorem 8, we compute the ECP of .
Theorem 8. Let and be graphs. Then,
Proof. Let be the root vertex of , having order and size . Also note that , , , for all and . By using Theorem 7, we have
Example 5. For the bottleneck graph , .
Example 6. If we use the vertices , , in the construction of a bridge graph, then it is obtained by connecting the vertices and of by a link for all and is represented by . For and , we can define . In particular, let with and be a type of bridge graphs. Then, we have , , and . The graphs , , and are depicted in Figure 5. By using Theorem 8, we gain the following results:
5. Conclusions
The numerical description of chemical structures with graph invariants is a valuable graph theory application. This characterization may be in the form of spectra, polynomials, molecular, or atomic topological indices. It is also feasible to specify graphs by matrices. A well-known example of such matrices is an adjacency matrix. However, the characterization of graphs by polynomials is a new line of research in modern graph theory. This paper is an effort in this direction, through which the ECP for some product graphs is illustrated by graph structure analysis and a mathematical derivation method.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research was supported by UPAR Grant of United Arab Emirates University (UAEU), Al Ain, UAE, via Grant nos. G00002590 and G00003271.