# Distance-Based Polynomials and Topological Indices for Hierarchical Hypercube Networks

Authors: Tingmei Gao; Iftikhar Ahmed
Journal: Journal of Mathematics
 (2021)

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## 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.


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## Body

## 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 hypercubeHHC [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 HHC is scalable, that is, it can embed HHCs of lower dimensions. Malluhi and Bayoumi [21] introduced the hierarchical hypercube network of order nn−HHC. The structure of an n−HHC consists of three levels of hierarchy. At the lowest level of hierarchy, there is a pool of 2n nodes. These 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. The set of the S-cubes constitutes the second level of hierarchy [23].Being a hierarchical structure, theHHC 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. The former property implies fast communication, and the latter implies easy mapping of operations from HC to HHC. The HHC can emulate the hypercube for a large class of problems (divide and conquer), without a significant increase in processing time. The HHC can embed rings and HHCs of lower dimension. In addition, the HHC embeds the cube connected cycle CCC [26–28]. As a result, the performance of HHC is in the worst case equivalent to the performance of the CCC [29–34]. The number of vertices and edges in HHC−1 is 16a+16 and 24a+20, respectively, where a is a natural number. The 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, respectively.Figure 1
The hierarchical hypercube networkHHC−12×2.Figure 2
The hierarchical hypercube networkHHC−22×2.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 graphG 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 u and v is denoted by du,v and is the length of the shortest path between u and v. The diameter of a graph is the maximum distance between any two vertices of G. 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 graphG, the Hosoya polynomial is defined as(1)HG,x=12∑y∈VG∑z∈VGxdy,z,where dy,z represents the distance between the vertices y and z.Definition 2.
(Wiener index [37]). The Wiener index for a simple connected graph G is denoted by WG and is defined as the sum of distances between all pairs of vertices in G, i.e.,(2)WG=12∑y∈VG∑z∈VGdy,z.
It can be observed that the Wiener index is the first-order derivative of the Hosoya polynomial atx=1.(3)WG=dHGdx|x=1.Definition 3.
(modified Wiener index). For a simple connected graphG, the modified Wiener index is denoted by WλG and is defined as(4)WλG=12∑y∈VG∑z∈VGdy,zλ,where λ is any positive integer.Definition 4.
(hyper-Wiener index). For a simple connected graphG, the hyper-Wiener index is denoted by WWG and is defined as(5)WWG=12∑y∈VG∑z∈VGdy,z+dy,z2.
The 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 byWWλG. For a simple connected graph G, the modified Wiener index is defined as(6)WWλG=12∑y∈VG∑z∈VGdy,zλ+dy,z2λ,where λ is any positive integer.Definition 6.
(Harary polynomial). The Harary polynomial for a simple connected graphG is denoted by hG and is defined as(7)hG=∑y∈VG∑z∈VG1dy,zxdy,z.Definition 7.
(generalized Harary index). The generalized Harary index for a simple connected graphG is denoted by htG and is defined as(8)htG=∑y∈VG∑z∈VG1dy,z+t,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.Definition 8.
(multiplicative Wiener index). The multiplicative Wiener index for a simple connected graphG is denoted by πG and is defined as(9)πG=∏y,z⊆VGdy,z.


## 3. Methodology

With the aid of distance matrix of a graphG, we can evaluate Hosoya polynomial. To calculate the Hosoya polynomial, we must calculate the number of pairs of vertices at distance 1,2,3,…,diaG, where diaG=maxdu,v;u,v∈VG. Mathematical induction will be used for the above cause. The usual vision of Hosoya polynomial is given below, where d represents the maximum distance in graph.(10)HG;x=a0nx0+a1nx1+a2nx2+⋯+adnxd.


## 4. Distance-Based Polynomials and Indices for Hierarchical Hypercube Networks

This section consists of two subsections. In Section4.1, 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 Theorem1, we give Hosoya polynomial of HHC-1.Theorem 1.
Forn≥3, the Hosoya polynomial of HHC−1 is(11)HHHC−1;x=20+24nx+24+40nx2+24+64nx3+4+100nx4+−28+124nx5+−80+132nx6+−152+132nx7+∑8≤m≤2n+2837−8m+16nxm+108x2n+3+60x2n+4+20x2n+5.Proof.
To prove this result, we need to calculateamn = 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 Table1, we have(12)a1n=20+24n,a2n=24+40n,a3n=24+64n,a4n=4+100n,a5n=−28+124n,a6n=−80+132n,a7n=−152+132n.
The remaining proof is divided into the following two main cases.Case 1. When m≡0mod2 and 8<m≤2n+2.It can be observed from Table2 that(13)a83=168,a84=296,a85=424,a86=552,a87=680,a88=808.Now, we can deduce that(14)a8n=40+128n−2.In a similar fashion, we have(15)a104=168,a105=296,a106=424,a107=552,a108=680,a109=808.It implies that(16)a10n=40+128n−3.In a similar fashion, we infer(17)a12n=40+128n−4,a14n=40+128n−5,a16n=40+128n−6,⋯.Now, we have(18)amn=40+128n−m−42=837−8m+16n.Case 2. When m≡1mod2 and 9<m≤2n+1.It can be observed from Table2 that(19)a94=232,a95=360,a96=488,a97=616,a98=744,a99=872.Now, we can deduce that(20)a9n=104+128n−3.By means of the same trick, we obtain(21)a115=232,a116=360,a117=488,a118=616,a119=744,a1110=872,which reveal that(22)a11n=104+128n−4.With a similar approach, we get(23)a13n=104+128n−5,a15n=104+128n−6,a17n=104+128n−7,⋯.Hence, we have(24)amn=104+128n−m−32amn=837−8m+16n.One can see that in both cases, we get the same result, so we can write for8<m≤2n+1 as(25)amn=837−8m+16n,and the remaining last three distances having fixed values are(26)a2n+3n=108,a2n+4n=60,a2n+5n=20.By what have been mentioned above and using definition of Hosoya polynomial, we arrive at our desired result.Table 1
Pair of vertices in HHC-1 at different distances.
mn345678...192116140164188212...2144184224264304344...3216280344408472536...4304404504604704804...5344468592716840964...6316448580712844976...7244376508640772904...Table 2
Pair of vertices in HHC-1 at different distances.
mn345678...8168296424552680808...9—232360488616744...10—168296424552680...11——232360488616...12——168296424552...13———232360488...14———168296424...15————232360...16————168296...17—————232...18—————168...19——————...20———————..21————————.In Theorem2, we give Harary polynomial for HHC-1.Theorem 2.
Forn≥3, the Harary polynomial of HHC−1 is(27)hHHC−1;x=20+24nx+24+40n2x2+24+64n3x3+4+100n4x4+−28+124n5x5+−80+132n6x6+−152+132n7x7+∑8≤m≤2n+2837−8m+16nmxm+1082n+3x2n+3+602n+4x2n+4+202n+5x2n+5.Proof.
The proof of this result is easy to follow by using information given in Theorem1 and definition of Harary polynomial.
In Theorem3, we give modified Wiener index, modified hyper-Wiener index, generalized Harary index, and multiplicative Wiener index of HHC-1.Theorem 3.
ForHHC−1, we have(1)
The modified Wiener index:(28)WλHHC−1=20+24n1λ+24+40n2λ+24+64n3λ+4+100n4λ+−28+124n5λ+−80+132n6λ+−152+132n7λ+∑8≤m≤2n+2837−8m+16nmλ+1082n+3λ+602n+4λ+202n+5λ.(2)
The modified hyper-Wiener index:(29)WWλHHC−1=20+24n1λ+12λ+24+40n2λ+22λ+24+64n3λ+32λ+4+100n4λ+42λ+−28+124n5λ+52λ+−80+132n6λ+62λ+−152+132n7λ+72λ+∑8≤m≤2n+2837−8m+16nmλ+m2λ+1082n+3λ+2n+32λ+602n+4λ+2n+42λ+202n+5λ+2n+52λ.(3)
The generalized Harary index:(30)htHHC−1=20+24n1+t+24+40n2+t+24+64n3+t+4+100n4+t+−28+124n5+t+−80+132n6+t+−152+132n7+t+∑8≤m≤2n+2837−8m+16n1m+t+1082n+3+t+602n+4+t+202n+5+t.(4)
The multiplicative Wiener index:(31)πHHC−1=120+24n×224+40n×324+64n×44+100n×5−28+124n×6−80+132n×7−152+132n×∏8<m≤2n+2m837−8m+16n×2n+3108×2n+460×2n+520.From Theorem3, we get the following results immediately.Corollary 1.
ForHHC−1, we have(32)WHHC−1=256n33+592n2+2696n3+376.Proof.
This result can be easily established by takingλ=1 in (1) of Theorem 3.Corollary 2.
ForHHC−1, we have(33)WWHHC−1=256n43+2624n33+10544n23+14056n3+1816.Proof.
This result can be easily established by takingλ=1 in (2) of Theorem 3.Corollary 3.
ForHHC−1, we have(34)HHHC−1=15599n100+634793176116977118014398509481984+∑8≤m≤2n+2837−8m+16nm+1082n+3+202n+5+30n+2.Proof.
This result can be easily established by takingt=1 in (3) of Theorem 3.


### 4.2. Distance-Based Polynomials and Indices for Hierarchical Hypercube Network HHC-2

In Theorem4, we give Hosoya polynomial for HHC-2.Theorem 4.
ForHHC−2, the Hosoya polynomial is(35)HHHC−2;x=28+32nx+24+48nx2+36+96nx3+−4+124nx4+−60+128nx5+−120+132nx6+∑m≡0mod2,8≤m≤2n+2833−8m+16nxm+∑m≡1mod2,7≤m≤2n+1−64m−22+nxm+68x2n+3+32x2n+4+4x2n+5.Proof.
To prove this result, we have to calculateamn where m=1,2,3,…,2n+5. Here amn 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 Table3, we have(36)a1n=28+32n,a2n=24+48n,a3n=36+96n,a4n=−4+124n,a5n=−60+128n,a6n=−120+132n.
The remaining proof is divided into the following two main cases.Case 1. When m≡0mod2 and 8<m≤2n+2.It can be observed from Table4 that(37)a83=136,a84=264,a85=392,a86=520,a87=648,a88=776.Now, we can deduce that(38)a8n=8+128n−2.In a similar fashion, we have(39)a104=136,a105=264,a106=392,a107=520,a108=648,a109=776.It implies that(40)a10n=8+128n−3.In a similar fashion, we infer(41)a12n=8+128n−4,a14n=8+128n−5,a16n=8+128n−6,⋯,which yield(42)amn=8+128n−m−42=833−8m+16n.Case 2. When m≡1mod2 and 7<m≤2n+1.It can be observed from Table4 that(43)a73=192,a74=320,a75=448,a76=576,a77=704,a78=832.Now, we can deduce that(44)a7n=64+128n−2.So, we obtain(45)a94=192,a95=320,a96=448,a97=576,a98=704,a99=832,which reveal that(46)a9n=64+128n−3.Also, we get(47)a13n=64+128n−4,a15n=64+128n−5,a17n=64+128n−6,⋯.Hence, we have(48)amn=64+128n−m−32=−64m−22+n,and the remaining last three distances having fixed values are(49)a2n+3n=68,a2n+4n=32,a2n+5n=4.By what have been mentioned above and using the definition of Hosoya polynomial, we arrive at our desired result.Table 3
Number of pair of vertices in HHC-2 at different distances.
mn234567...192124156188220252...2120168216264312360...3228324420516612708...4244368492616740864...5196324452580708836...6144276408540672804...Table 4
Number of pair of vertices of HHC-2 at different distances.
mn345678...7192320448576704832...8136264392520648776...9—192320448576704...10—136264392520648...11——192320448576...12——136264392520...13———192320448...14———136264392...15————192320...16————136264...17—————192...18—————136...19——————...20———————..21————————.In Theorem5, we give Harary polynomial for HHC-2.Theorem 5.
ForHHC−2, the Harary polynomial is(50)hHHC−2;x=28+32nx+24+48n2x2+36+96n3x3+36+96n4x4+−60+128n5x5+−120+132n6x6+∑8≤m≤2n+2833−8m+16nmxm+∑7≤m≤2n+1−64m−22+nmxm+682n+3x2n+3+322n+4x2n+4+42n+5x2n+5.Proof.
The proof of this theorem is straightforward from the facts specified in Theorem4 and by definition of Harary polynomial.In Theorem6, we give modified Wiener index, modified hyper-Wiener index, generalized Harary index, and multiplicative Wiener index for HHC-2.Theorem 6.
ForHHC−2, we have(1)
The modified Wiener index:(51)WλHHC−2=28+32n1λ+24+48n2λ+36+96n3λ+36+96n4λ+−60+128n5λ+−120+132n6λ+∑m≡0mod2,8≤m≤2n+2833−8m+16nmλ+∑m≡1mod2,7≤m≤2n+1−64m−22+nmλ+682n+3λ+322n+4λ+42n+5λ.(2)
The modified hyper-Wiener index:(52)WWλHHC−2=28+32n1λ+12λ+24+48n2λ+22λ+36+96n3λ+32λ+36+96n4λ+42λ+−60+128n5λ+52λ+−120+132n6λ+62λ+∑m≡0mod2,8≤m≤2n+2833−8m+16nmλ+m2λ+∑m≡1mod2,7≤m≤2n+1−64m−22+nmλ+m2λ+682n+3λ+2n+32λ+322n+4λ+2n+42λ+42n+5λ+2n+52λ.(3)
The generalized Harary index:(53)htHHC−2=28+32n1+t+24+48n2+t+36+96n3+t+36+96n4+t+−60+128n5+t+−120+132n6+t+∑m≡0mod2,8≤m≤2n+2833−8m+16n1m+t+∑m≡1mod2,7≤m≤2n+1−64m−22+n1m+t+1082n+3+t+602n+4+t+202n+5+t.(4)
The multiplicative Wiener index:(54)πHHC−2=128+32n×224+48n×336+96n×436+96n×5−60+128n×6−120+132n×∏m≡0mod2,8<m≤2n+2m833−8m+16n×∏m≡1mod2,7<m≤2n+1m−64m−22+n2n+368×2n+432×2n+54.From Theorem6, we get the following results immediately.Corollary 4.
ForHHC−2, we have(55)WHHC−2=256n33+520n2+2288n3+316.Proof.
This result can be easily established by takingλ=1 in (1) of Theorem 6.Corollary 5.
ForHHC−2, we have(56)WWHHC−2=256n43+2336n33+8264n23+10480n3+1320.Proof.
This result can be easily established by takingλ=1 in (2) of Theorem 6.Corollary 6.
ForHHC−2, we have(57)HHHC−2=798n5+29+∑m≡0mod2,8≤m≤2n+2833−8m+16n1m+∑m≡1mod2,7≤m≤2n+1−64m−22+n1m+1082n+3+202n+5+30n+2.Proof.
This result can be easily established by takingt=1 in (3) of Theorem 6.


## 4.1. Distance-Based Polynomials and Indices for Hierarchical Hypercube Network HHC-1

In Theorem1, we give Hosoya polynomial of HHC-1.Theorem 1.
Forn≥3, the Hosoya polynomial of HHC−1 is(11)HHHC−1;x=20+24nx+24+40nx2+24+64nx3+4+100nx4+−28+124nx5+−80+132nx6+−152+132nx7+∑8≤m≤2n+2837−8m+16nxm+108x2n+3+60x2n+4+20x2n+5.Proof.
To prove this result, we need to calculateamn = 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 Table1, we have(12)a1n=20+24n,a2n=24+40n,a3n=24+64n,a4n=4+100n,a5n=−28+124n,a6n=−80+132n,a7n=−152+132n.
The remaining proof is divided into the following two main cases.Case 1. When m≡0mod2 and 8<m≤2n+2.It can be observed from Table2 that(13)a83=168,a84=296,a85=424,a86=552,a87=680,a88=808.Now, we can deduce that(14)a8n=40+128n−2.In a similar fashion, we have(15)a104=168,a105=296,a106=424,a107=552,a108=680,a109=808.It implies that(16)a10n=40+128n−3.In a similar fashion, we infer(17)a12n=40+128n−4,a14n=40+128n−5,a16n=40+128n−6,⋯.Now, we have(18)amn=40+128n−m−42=837−8m+16n.Case 2. When m≡1mod2 and 9<m≤2n+1.It can be observed from Table2 that(19)a94=232,a95=360,a96=488,a97=616,a98=744,a99=872.Now, we can deduce that(20)a9n=104+128n−3.By means of the same trick, we obtain(21)a115=232,a116=360,a117=488,a118=616,a119=744,a1110=872,which reveal that(22)a11n=104+128n−4.With a similar approach, we get(23)a13n=104+128n−5,a15n=104+128n−6,a17n=104+128n−7,⋯.Hence, we have(24)amn=104+128n−m−32amn=837−8m+16n.One can see that in both cases, we get the same result, so we can write for8<m≤2n+1 as(25)amn=837−8m+16n,and the remaining last three distances having fixed values are(26)a2n+3n=108,a2n+4n=60,a2n+5n=20.By what have been mentioned above and using definition of Hosoya polynomial, we arrive at our desired result.Table 1
Pair of vertices in HHC-1 at different distances.
mn345678...192116140164188212...2144184224264304344...3216280344408472536...4304404504604704804...5344468592716840964...6316448580712844976...7244376508640772904...Table 2
Pair of vertices in HHC-1 at different distances.
mn345678...8168296424552680808...9—232360488616744...10—168296424552680...11——232360488616...12——168296424552...13———232360488...14———168296424...15————232360...16————168296...17—————232...18—————168...19——————...20———————..21————————.In Theorem2, we give Harary polynomial for HHC-1.Theorem 2.
Forn≥3, the Harary polynomial of HHC−1 is(27)hHHC−1;x=20+24nx+24+40n2x2+24+64n3x3+4+100n4x4+−28+124n5x5+−80+132n6x6+−152+132n7x7+∑8≤m≤2n+2837−8m+16nmxm+1082n+3x2n+3+602n+4x2n+4+202n+5x2n+5.Proof.
The proof of this result is easy to follow by using information given in Theorem1 and definition of Harary polynomial.
In Theorem3, we give modified Wiener index, modified hyper-Wiener index, generalized Harary index, and multiplicative Wiener index of HHC-1.Theorem 3.
ForHHC−1, we have(1)
The modified Wiener index:(28)WλHHC−1=20+24n1λ+24+40n2λ+24+64n3λ+4+100n4λ+−28+124n5λ+−80+132n6λ+−152+132n7λ+∑8≤m≤2n+2837−8m+16nmλ+1082n+3λ+602n+4λ+202n+5λ.(2)
The modified hyper-Wiener index:(29)WWλHHC−1=20+24n1λ+12λ+24+40n2λ+22λ+24+64n3λ+32λ+4+100n4λ+42λ+−28+124n5λ+52λ+−80+132n6λ+62λ+−152+132n7λ+72λ+∑8≤m≤2n+2837−8m+16nmλ+m2λ+1082n+3λ+2n+32λ+602n+4λ+2n+42λ+202n+5λ+2n+52λ.(3)
The generalized Harary index:(30)htHHC−1=20+24n1+t+24+40n2+t+24+64n3+t+4+100n4+t+−28+124n5+t+−80+132n6+t+−152+132n7+t+∑8≤m≤2n+2837−8m+16n1m+t+1082n+3+t+602n+4+t+202n+5+t.(4)
The multiplicative Wiener index:(31)πHHC−1=120+24n×224+40n×324+64n×44+100n×5−28+124n×6−80+132n×7−152+132n×∏8<m≤2n+2m837−8m+16n×2n+3108×2n+460×2n+520.From Theorem3, we get the following results immediately.Corollary 1.
ForHHC−1, we have(32)WHHC−1=256n33+592n2+2696n3+376.Proof.
This result can be easily established by takingλ=1 in (1) of Theorem 3.Corollary 2.
ForHHC−1, we have(33)WWHHC−1=256n43+2624n33+10544n23+14056n3+1816.Proof.
This result can be easily established by takingλ=1 in (2) of Theorem 3.Corollary 3.
ForHHC−1, we have(34)HHHC−1=15599n100+634793176116977118014398509481984+∑8≤m≤2n+2837−8m+16nm+1082n+3+202n+5+30n+2.Proof.
This result can be easily established by takingt=1 in (3) of Theorem 3.


## 4.2. Distance-Based Polynomials and Indices for Hierarchical Hypercube Network HHC-2

In Theorem4, we give Hosoya polynomial for HHC-2.Theorem 4.
ForHHC−2, the Hosoya polynomial is(35)HHHC−2;x=28+32nx+24+48nx2+36+96nx3+−4+124nx4+−60+128nx5+−120+132nx6+∑m≡0mod2,8≤m≤2n+2833−8m+16nxm+∑m≡1mod2,7≤m≤2n+1−64m−22+nxm+68x2n+3+32x2n+4+4x2n+5.Proof.
To prove this result, we have to calculateamn where m=1,2,3,…,2n+5. Here amn 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 Table3, we have(36)a1n=28+32n,a2n=24+48n,a3n=36+96n,a4n=−4+124n,a5n=−60+128n,a6n=−120+132n.
The remaining proof is divided into the following two main cases.Case 1. When m≡0mod2 and 8<m≤2n+2.It can be observed from Table4 that(37)a83=136,a84=264,a85=392,a86=520,a87=648,a88=776.Now, we can deduce that(38)a8n=8+128n−2.In a similar fashion, we have(39)a104=136,a105=264,a106=392,a107=520,a108=648,a109=776.It implies that(40)a10n=8+128n−3.In a similar fashion, we infer(41)a12n=8+128n−4,a14n=8+128n−5,a16n=8+128n−6,⋯,which yield(42)amn=8+128n−m−42=833−8m+16n.Case 2. When m≡1mod2 and 7<m≤2n+1.It can be observed from Table4 that(43)a73=192,a74=320,a75=448,a76=576,a77=704,a78=832.Now, we can deduce that(44)a7n=64+128n−2.So, we obtain(45)a94=192,a95=320,a96=448,a97=576,a98=704,a99=832,which reveal that(46)a9n=64+128n−3.Also, we get(47)a13n=64+128n−4,a15n=64+128n−5,a17n=64+128n−6,⋯.Hence, we have(48)amn=64+128n−m−32=−64m−22+n,and the remaining last three distances having fixed values are(49)a2n+3n=68,a2n+4n=32,a2n+5n=4.By what have been mentioned above and using the definition of Hosoya polynomial, we arrive at our desired result.Table 3
Number of pair of vertices in HHC-2 at different distances.
mn234567...192124156188220252...2120168216264312360...3228324420516612708...4244368492616740864...5196324452580708836...6144276408540672804...Table 4
Number of pair of vertices of HHC-2 at different distances.
mn345678...7192320448576704832...8136264392520648776...9—192320448576704...10—136264392520648...11——192320448576...12——136264392520...13———192320448...14———136264392...15————192320...16————136264...17—————192...18—————136...19——————...20———————..21————————.In Theorem5, we give Harary polynomial for HHC-2.Theorem 5.
ForHHC−2, the Harary polynomial is(50)hHHC−2;x=28+32nx+24+48n2x2+36+96n3x3+36+96n4x4+−60+128n5x5+−120+132n6x6+∑8≤m≤2n+2833−8m+16nmxm+∑7≤m≤2n+1−64m−22+nmxm+682n+3x2n+3+322n+4x2n+4+42n+5x2n+5.Proof.
The proof of this theorem is straightforward from the facts specified in Theorem4 and by definition of Harary polynomial.In Theorem6, we give modified Wiener index, modified hyper-Wiener index, generalized Harary index, and multiplicative Wiener index for HHC-2.Theorem 6.
ForHHC−2, we have(1)
The modified Wiener index:(51)WλHHC−2=28+32n1λ+24+48n2λ+36+96n3λ+36+96n4λ+−60+128n5λ+−120+132n6λ+∑m≡0mod2,8≤m≤2n+2833−8m+16nmλ+∑m≡1mod2,7≤m≤2n+1−64m−22+nmλ+682n+3λ+322n+4λ+42n+5λ.(2)
The modified hyper-Wiener index:(52)WWλHHC−2=28+32n1λ+12λ+24+48n2λ+22λ+36+96n3λ+32λ+36+96n4λ+42λ+−60+128n5λ+52λ+−120+132n6λ+62λ+∑m≡0mod2,8≤m≤2n+2833−8m+16nmλ+m2λ+∑m≡1mod2,7≤m≤2n+1−64m−22+nmλ+m2λ+682n+3λ+2n+32λ+322n+4λ+2n+42λ+42n+5λ+2n+52λ.(3)
The generalized Harary index:(53)htHHC−2=28+32n1+t+24+48n2+t+36+96n3+t+36+96n4+t+−60+128n5+t+−120+132n6+t+∑m≡0mod2,8≤m≤2n+2833−8m+16n1m+t+∑m≡1mod2,7≤m≤2n+1−64m−22+n1m+t+1082n+3+t+602n+4+t+202n+5+t.(4)
The multiplicative Wiener index:(54)πHHC−2=128+32n×224+48n×336+96n×436+96n×5−60+128n×6−120+132n×∏m≡0mod2,8<m≤2n+2m833−8m+16n×∏m≡1mod2,7<m≤2n+1m−64m−22+n2n+368×2n+432×2n+54.From Theorem6, we get the following results immediately.Corollary 4.
ForHHC−2, we have(55)WHHC−2=256n33+520n2+2288n3+316.Proof.
This result can be easily established by takingλ=1 in (1) of Theorem 6.Corollary 5.
ForHHC−2, we have(56)WWHHC−2=256n43+2336n33+8264n23+10480n3+1320.Proof.
This result can be easily established by takingλ=1 in (2) of Theorem 6.Corollary 6.
ForHHC−2, we have(57)HHHC−2=798n5+29+∑m≡0mod2,8≤m≤2n+2833−8m+16n1m+∑m≡1mod2,7≤m≤2n+1−64m−22+n1m+1082n+3+202n+5+30n+2.Proof.
This result can be easily established by takingt=1 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.



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