Distance-Based Polynomials and Topological Indices for Hierarchical Hypercube Networks

Topological indices are the numbers associated with the graphs of chemical compounds/networks that help us to understand their properties.-e 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.


Introduction
e field of mathematics which deals with the problems of chemistry mathematically is mathematical chemistry. e 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][2][3] can be obtained from the indices.
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][15][16][17][18]. is paper concerns with the topological indices of hierarchical hypercube networks. Interconnection networks have a pivotal role in the execution of parallel systems. is paper studies the interconnection topology that is called the hierarchical hypercube (HHC) [19]. is 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 HHC is scalable, that is, it can embed HHCs of lower dimensions. Malluhi and Bayoumi [21] introduced the hierarchical hypercube network of order n (n − HHC). e structure of an n − HHC consists of three levels of hierarchy. At the lowest level of hierarchy, there is a pool of 2n nodes. ese nodes are grouped into clusters of 2m nodes each, and the nodes in each cluster are interconnected to form an m − cube called the Son-cube or the S-cube. e set of the S-cubes constitutes the second level of hierarchy [23].
Being a hierarchical structure, the HHC 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 HHC structure are logarithmic diameter and a topology inherited from, and closely related to, the hypercube topology. e former property implies fast communication, and the latter implies easy mapping of operations from HC to HHC. e HHC can emulate the hypercube for a large class of problems (divide and conquer), without a significant increase in processing time. e HHC can embed rings and HHCs of lower dimension. In addition, the HHC embeds the cube connected cycle (CCC) [26][27][28]. As a result, the performance of HHC is in the worst case equivalent to the performance of the CCC [29][30][31][32][33][34]. e number of vertices and edges in (HHC − 1) is 16a + 16 and 24a + 20, respectively, where a is a natural number. e number of vertices and edges in (HHC − 2) is 16a + 16 and 32a + 28, respectively. (HHC − 1) and (HHC − 2) are shown in Figures 1 and 2

Preliminaries
In this section, we give the definitions and known results that are used in proving main results of this paper. A graph G is simple if it has no loop or multiple edges and is connected if there is a path between every two vertices of it. e distance between any two vertices u and v is denoted by d (u, v) and is the length of the shortest path between u and v. e diameter of a graph is the maximum distance between any two vertices of G. e notions that are used in this paper but not defined can be found in [35,36].
For a simple connected graph G, the Hosoya polynomial is defined as where d(y, z) represents the distance between the vertices y and z.
Definition 2 (Wiener index [37]). e Wiener index for a simple connected graph G is denoted by W(G) and is defined as the sum of distances between all pairs of vertices in G, i.e., It can be observed that the Wiener index is the first-order derivative of the Hosoya polynomial at x � 1.
Definition 3 (modified Wiener index). For a simple connected graph G, the modified Wiener index is denoted by W λ (G) and is defined as where λ is any positive integer.
Definition 4 (hyper-Wiener index). For a simple connected graph G, the hyper-Wiener index is denoted by WW(G) and is defined as e 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 WW λ (G). For a simple connected graph G, the modified Wiener index is defined as where λ is any positive integer.
Definition 6 (Harary polynomial). e Harary polynomial for a simple connected graph G is denoted by h(G) and is defined as  e generalized Harary index for a simple connected graph G is denoted by h t (G) and is defined as where t � 1, 2, 3, 4, . . ..
For detailed study on Harary polynomial and Harary index, we refer to the readers [39,40] and references therein.
e multiplicative Wiener index for a simple connected graph G is denoted by π(G) and is defined as

Methodology
With the aid of distance matrix of a graph G, we can evaluate Hosoya polynomial. To calculate the Hosoya polynomial, we must calculate the number of pairs of vertices at distance Mathematical induction will be used for the above cause. e usual vision of Hosoya polynomial is given below, where d represents the maximum distance in graph.

Distance-Based Polynomials and Indices for
Hierarchical Hypercube Networks is 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.
Proof. To prove this result, we need to calculate |a m (n)| � number of pair of vertices at distance m, where m � 1, 2, 3, . . . , 2n + 5. 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 have e remaining proof is divided into the following two main cases.
It can be observed from Table 2 that Now, we can deduce that In a similar fashion, we have In a similar fashion, we infer Now, we have Case 2. When m ≡ 1(mod2) and 9 < m ≤ 2n + 1.
It can be observed from Table 2 that Now, we can deduce that a 9 (n) � 104 + 128(n − 3).

(26)
By what have been mentioned above and using definition of Hosoya polynomial, we arrive at our desired result.

Theorem 2. For n ≥ 3, the Harary polynomial of HHC
Proof. e proof of this result is easy to follow by using information given in eorem 1 and definition of Harary polynomial.
In eorem 3, we give modified Wiener index, modified hyper-Wiener index, generalized Harary index, and multiplicative Wiener index of HHC-1.
is result can be easily established by taking t � 1 in (3) of eorem 3.
e remaining proof is divided into the following two main cases.
It can be observed from Table 4 Now, we can deduce that In a similar fashion, we have It implies that In a similar fashion, we infer Case 2. When m ≡ 1(mod2) and 7 < m ≤ 2n + 1.
It can be observed from Table 4 which reveal that Also, we get By what have been mentioned above and using the definition of Hosoya polynomial, we arrive at our desired result.

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 structureactivity 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
e data used to support the findings of this study are included within the article.