Abstract
Topological indices are the numbers associated with the graphs of chemical compounds/networks that help us to understand their properties. The aim of this paper is to compute topological indices for the hierarchical hypercube networks. We computed Hosoya polynomials, Harary polynomials, Wiener index, modified Wiener index, hyper-Wiener index, Harary index, generalized Harary index, and multiplicative Wiener index for hierarchical hypercube networks. Our results can help to understand topology of hierarchical hypercube networks and are useful to enhance the ability of these networks. Our results can also be used to solve integral equations.
1. Introduction
The field of mathematics which deals with the problems of chemistry mathematically is mathematical chemistry. The topology of chemical structures, for example, topological labels or indices, is investigated in chemical graph theory. Actually, topological indices are real numbers associated with graph of chemical compounds and are useful in quantitative structure-property relationships and quantitative structure-activity relationships. Topological indices predict some important properties of chemical structures even without using lab, for example, boiling point, viscosity, radius of gyration, and so on [1–3] can be obtained from the indices.
Just like topological indices, polynomials also have significant applications in chemistry, for example, Hosoya polynomial [4] plays a vital role in calculation of distance-based topological indices. Like Hosoya polynomial, M-polynomial [5] plays the same role in calculation of many degree-based TIs [6–12].
Wiener defined the first topological index when he was examining boiling point of paraffins [13]. Consequently, Wiener set up the skeleton of topological indices [14–18].
This paper concerns with the topological indices of hierarchical hypercube networks. Interconnection networks have a pivotal role in the execution of parallel systems. This paper studies the interconnection topology that is called the hierarchical hypercube [19]. This topology is suitable for extensively parallel systems with many number of processors. An appealing property of this network is the low number of connections per processor, which enhances the VLSI design and fabrication of the system [20, 21]. Other alluring features include symmetry and logarithmic diameter, which imply easy and fast algorithms for communication [22]. Moreover, the is scalable, that is, it can embed s of lower dimensions. Malluhi and Bayoumi [21] introduced the hierarchical hypercube network of order n. The structure of an consists of three levels of hierarchy. At the lowest level of hierarchy, there is a pool of nodes. These nodes are grouped into clusters of nodes each, and the nodes in each cluster are interconnected to form an called the Son-cube or the S-cube. The set of the S-cubes constitutes the second level of hierarchy [23].
Being a hierarchical structure, the bears the advantages usually gained by hierarchy [24]. In general, hierarchy is a useful means for modular design. In addition, hierarchical structures are capable of exploiting the locality of reference (communication), and they are fault tolerant [25]. Other attractive properties of the structure are logarithmic diameter and a topology inherited from, and closely related to, the hypercube topology. The former property implies fast communication, and the latter implies easy mapping of operations from to . The can emulate the hypercube for a large class of problems (divide and conquer), without a significant increase in processing time. The can embed rings and s of lower dimension. In addition, the embeds the cube connected cycle [26–28]. As a result, the performance of is in the worst case equivalent to the performance of the [29–34]. The number of vertices and edges in is and , respectively, where is a natural number. The number of vertices and edges in is and , respectively. and are shown in Figures 1 and 2, respectively.
In this paper, we computed Hosoya polynomials, Harary polynomials, Wiener index, modified Wiener index, hyper-Wiener index, Harary index, generalized Harary index, and multiplicative Wiener index for hierarchical hypercube networks.
2. Preliminaries
In this section, we give the definitions and known results that are used in proving main results of this paper. A graph is simple if it has no loop or multiple edges and is connected if there is a path between every two vertices of it. The distance between any two vertices and is denoted by and is the length of the shortest path between and . The diameter of a graph is the maximum distance between any two vertices of . The notions that are used in this paper but not defined can be found in [35, 36].
Definition 1. (Hosoya polynomial [16]).
For a simple connected graph , the Hosoya polynomial is defined aswhere represents the distance between the vertices and .
Definition 2. (Wiener index [37]). The Wiener index for a simple connected graph is denoted by and is defined as the sum of distances between all pairs of vertices in , i.e.,It can be observed that the Wiener index is the first-order derivative of the Hosoya polynomial at .
Definition 3. (modified Wiener index). For a simple connected graph , the modified Wiener index is denoted by and is defined aswhere is any positive integer.
Definition 4. (hyper-Wiener index). For a simple connected graph , the hyper-Wiener index is denoted by and is defined asThe hyper-Wiener index (HWI) was introduced by Randić [38] and is used to forecast physical chemistry characteristics of organic compounds.
Definition 5. (modified hyper-Wiener index). Another variant of Wiener index is hyper-Wiener index (MHWI) which is denoted by . For a simple connected graph , the modified Wiener index is defined aswhere is any positive integer.
Definition 6. (Harary polynomial). The Harary polynomial for a simple connected graph is denoted by and is defined as
Definition 7. (generalized Harary index). The generalized Harary index for a simple connected graph is denoted by and is defined aswhere .
For detailed study on Harary polynomial and Harary index, we refer to the readers [39, 40] and references therein.
Definition 8. (multiplicative Wiener index). The multiplicative Wiener index for a simple connected graph is denoted by and is defined as
3. Methodology
With the aid of distance matrix of a graph , we can evaluate Hosoya polynomial. To calculate the Hosoya polynomial, we must calculate the number of pairs of vertices at distance , where . Mathematical induction will be used for the above cause. The usual vision of Hosoya polynomial is given below, where represents the maximum distance in graph.
4. Distance-Based Polynomials and Indices for Hierarchical Hypercube Networks
This section consists of two subsections. In Section 4.1, we present results about HHC-1, and in Section 4.2, we present results about HHC-2.
4.1. Distance-Based Polynomials and Indices for Hierarchical Hypercube Network HHC-1
In Theorem 1, we give Hosoya polynomial of HHC-1.
Theorem 1. For , the Hosoya polynomial of is
Proof. To prove this result, we need to calculate = number of pair of vertices at distance m, where . Using Figure 1, the number of pair of vertices at different distances is computed and is listed in Tables 1 and 2.
Now, by using Table 1, we haveThe remaining proof is divided into the following two main cases. Case 1. When and . It can be observed from Table 2 that Now, we can deduce that In a similar fashion, we have It implies that In a similar fashion, we infer Now, we have Case 2. When and . It can be observed from Table 2 that Now, we can deduce that By means of the same trick, we obtain which reveal that With a similar approach, we get Hence, we have One can see that in both cases, we get the same result, so we can write for as and the remaining last three distances having fixed values are By what have been mentioned above and using definition of Hosoya polynomial, we arrive at our desired result.
In Theorem 2, we give Harary polynomial for HHC-1.
Theorem 2. For , the Harary polynomial of is
Proof. The proof of this result is easy to follow by using information given in Theorem 1 and definition of Harary polynomial.
In Theorem 3, we give modified Wiener index, modified hyper-Wiener index, generalized Harary index, and multiplicative Wiener index of HHC-1.
Theorem 3. For , we have(1)The modified Wiener index:(2)The modified hyper-Wiener index:(3)The generalized Harary index:(4)The multiplicative Wiener index:
From Theorem 3, we get the following results immediately.
Corollary 1. For , we have
Proof. This result can be easily established by taking in (1) of Theorem 3.
Corollary 2. For , we have
Proof. This result can be easily established by taking in (2) of Theorem 3.
Corollary 3. For , we have
Proof. This result can be easily established by taking in (3) of Theorem 3.
4.2. Distance-Based Polynomials and Indices for Hierarchical Hypercube Network HHC-2
In Theorem 4, we give Hosoya polynomial for HHC-2.
Theorem 4. For , the Hosoya polynomial is
Proof. To prove this result, we have to calculate where . Here is the same as in Theorem 1. From Figure 2, we can compute the number of pair of vertices in HHC-2 at different distances, which is given in Tables 3 and 4.
Now from Table 3, we haveThe remaining proof is divided into the following two main cases. Case 1. When and . It can be observed from Table 4 that Now, we can deduce that In a similar fashion, we have It implies that In a similar fashion, we infer which yield Case 2. When and . It can be observed from Table 4 that Now, we can deduce that So, we obtain which reveal that Also, we get Hence, we have and the remaining last three distances having fixed values are By what have been mentioned above and using the definition of Hosoya polynomial, we arrive at our desired result.
In Theorem 5, we give Harary polynomial for HHC-2.
Theorem 5. For , the Harary polynomial is
Proof. The proof of this theorem is straightforward from the facts specified in Theorem 4 and by definition of Harary polynomial.
In Theorem 6, we give modified Wiener index, modified hyper-Wiener index, generalized Harary index, and multiplicative Wiener index for HHC-2.
Theorem 6. For , we have(1)The modified Wiener index:(2)The modified hyper-Wiener index:(3)The generalized Harary index:(4)The multiplicative Wiener index:
From Theorem 6, we get the following results immediately.
Corollary 4. For , we have
Proof. This result can be easily established by taking in (1) of Theorem 6.
Corollary 5. For , we have
Proof. This result can be easily established by taking in (2) of Theorem 6.
Corollary 6. For , we have
Proof. This result can be easily established by taking in (3) of Theorem 6.
5. Conclusion
Topological indices can be applied in different fields of science, such as material science, arithmetic, informatics, biology, and so on. However, the most critical use of topological indices to date is in the nonexact quantitative structure-property relationships and quantitative structure-activity relationships. In this paper, we studied hierarchical hypercube networks. We computed distance-based polynomials and distance-based indices for these networks. In fact, we computed Hosoya polynomials, Harary polynomials, Wiener index, and different variants of Wiener indices for the studied networks. Our results can help in understanding the topology of hierarchical hypercube networks and can be used to solve integral equations.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
Tingmei Gao wrote the final version of this paper, analyzed the results, and arranged funding for this study. Iftikhar Ahmed proved the results and wrote the first version of this paper.