1. bookVolume 6 (2021): Issue 1 (January 2021)
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Computing Forgotten Topological Index of Extremal Cactus Chains

Online veröffentlicht: 31 Dec 2020
Seitenbereich: 439 - 446
Eingereicht: 07 Nov 2019
Akzeptiert: 17 Jul 2020
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Format
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01 Jan 2016
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2 Hefte pro Jahr
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Introduction

The topological indices expect a fundamental activity in synthetic graph hypothesis, especially in QSAR and QSPR assessments. Graph speculation has given a variety of significant mechanical assemblies to the theoretical physicists, for instance, the topological indices, topological systems and counting polynomials. The information on the substance constitution of iota is addressed by a sub-nuclear chart. The vertices and edges of sub-nuclear charts are contrasting with the atoms of the blends and substance bonds, separately. A portrayal of the structure or a condition of the particles is extraordinarily valuable in complex preliminaries. Topological lists and physico-invention possessions are used for the illustrating, pharmacologic, toxicological, characteristic and various possessions of engineered blends. A segment of the topological index that are most investigated in compound charts theory are Wiener index.

The topological index is addressed as a lone number that is partners the particular properties of an outline. The degree based topological index paly a fundamental occupation in substance graph speculation and particularly in theoretical science. The first and most analyzed record is the Wiener list that relies upon partition. Its old name is way record anyway later on it was renamed as Wiener file [32]. Let G be a basic and associated diagram with request n and size m. The level of a vertex is quantity of edges that are episode to it and meant by Θ(u). Zagreb files are two most seasoned most contemplated gradation built topological lists [33]. The first and second Zagreb lists are characterized as: M1(G)=vV(Θ(v))2=uvE(Θ(u)+Θ(v)),M2(G)=uvE[Θ(u)×Θ(v)]. \matrix{ {{M_1}(G) = \sum\limits_{v \in V} {{(\Theta (v))}^2} = \sum\limits_{uv \in E} (\Theta (u) + \Theta (v)),} \cr {{M_2}(G) = \sum\limits_{uv \in E} [\Theta (u) \times \Theta (v)].} \cr }

These two Zagreb records have been used and broke down to examination of sub-nuclear multifaceted nature, ZE- isomerism, chirality, and hetero-organizations. Progressively approximately their physico-manufactured solicitations and logical possessions canister be establish in [15, 16], independently.

Gutman et al. [14] decided the harsh formulae for the total π-electron vitality. One of them is the central Zagreb file and F– list was experienced on which this imperativeness depends, yet that was completely ignored. This rundown is used just in scarcely any business identified with the essential general Zagreb list and zeroth-request general Randić list. Furtula et al. [13] renamed this topological record as a disregarded topological list and they gained some fascinating results. The F-list is portrayed as: F(G)=vV(Θ(v))3. F(G) = \sum\limits_{v \in V} {(\Theta (v))^3}. The M1α(G) M_1^\alpha (G) is characterized as : M1α(G)=vVΘ(v)α=uvE[(Θ(u))(α1)+(Θ(v))(α1)]. M_1^\alpha (G) = \sum\limits_{v \in V} \Theta {(v)^\alpha } = \sum\limits_{uv \in E} [(\Theta (u{))^{(\alpha - 1)}} + {(\Theta (v))^{(\alpha - 1)}}]. Where α ≠ 0, α ∈ ℝ, α ≠ 1. Li et al. [22, 25] demarcated the Special wide-ranging Randić index as: Rα0(G)=uV(Θ(u))α, R_\alpha ^0(G) = \sum\limits_{u \in V} {(\Theta (u))^\alpha }, for all real α.

In the event that α = 3, at that point zeroth-request general Randić file is equivalent to the F-list. A few peak possessions of F-record for sub-atomic trees and approximately poorer and superior limits for the F-list were prearranged in [1]. Hosamani figured some upper limits for the F-list and its a few applications [19]. Nilanjan et al. [12] processed the specific recipes for the chart activities regarding the F-file. Yarahmaadi et al. [30, 31] processed the unconventional availability record of the chain hexagonal cacti.

Li et al. [23] decided the Hosoya polynomials of chains. A few properties of the prickly plant chains are given in [20, 24]. Xu et al. [27] processed the PI file for polyomino chains. Randić record and whole network file of polyomino chains are now concentrated in [28, 29]. The general Randic record of extremal polyomino chains decided in [3,4,5,6,7,8,9,10].

Results on Cactus Chains

In this segment, we figure the specific equations for the desert flora binds regarding the F-record and furthermore decided the extremal fastens as for the F-list. A prickly plant diagram is a straightforward associated chart in which nope edge fabrications in supplementary than individual cycle. Each square of a desert flora chart is whichever a cycle or an edge. On the off chance that all squares of a desert plant are patterns of a similar size m, at that point these sort of prickly plant charts are supposed to be m-uniform. The quantity of cycles in a m-uniform desert flora chain is baptized measurement of manacle. Besides, any m-uniform desert plant manacle of measurement more prominent than solitary has precisely two fatal cycles. Every residual cycle in chain are called inner cycles. More than 50 years prior, the desert flora charts were known as Husimi trees [18, 26].

Triangular Cactus Chains

A triangular desert flora is a diagram together tetragons are patterns of request 3 i.e., 3-unvarying prickly plant chains. In the triangular desert plant Tn, if each pattern of request 3 has all things considered twofold expurgated vertices and apiece expurgated vertex is pooled by accurately twofold patterns of request 3, at that point Tn is known as a chain three-cornered prickly plant. The quantity of pinnacles and quantity of authorities of fetter wedge-shaped prickly plant of extent n are 2n + 1 and 3n, separately. Clearly, all equivalent extent chain triangular prickly plants are isomorphic. A diagram T5 is appeared in Figure 1. Next, we process the F-record of triangular desert flora chain.

Fig. 1

The graphs T5 and W4.

Theorem 1

For n ≥ 2, the F-index is F(Tn) = 72n − 48.

The windmill diagram signified by Wn is the wedge-shaped desert plant with n trios that all the trios portion a vertex. The quantity of pinnacles and superiorities of windmill diagram Wn are 2n + 1 and 3n individually. A case of the windmill diagram is appeared in Figure 1. In the accompanying hypothesis, we process the F-record of windmill chart Wn.

Theorem 2

For n ≥ 1, the F-index is F(Wn) = 8n3 + 16n.

Square Cactus Chains

A Square Cactus Chains is a diagram with tetragons are C4 i.e., 4-undeviating. The quantity of vertices and the quantity of edges of the square prickly plant are 3n + 1 and 4n, separately. On the off chance that the twofold vertices are nearby in C4, at that point such tetragonal is an ortho- tetragons and is meant by Sn. In the event that twofold pinnacles are not immediate in C4, at that point such tetragons are christened para-tetragons and is indicated by Qn (These phrasings engaged after the hypothesis of benzenoid hydrocarbons [11]. The ortho and para restraints of tetragons are appeared in Figure 2.

Fig. 2

The ortho-chain S8 and para-chain Q6.

Theorem 3

For n ≥ 2, the F-index is F(Sn) = F(Qn) = 80n − 48.

Fig. 3

The ortho-chain, meta-chain and para chain.

Hexagonal Cactus Chains

A Hexagonal Cactus cawsers is a 6-undeviating prickly plant. The quantity of vertices and the edges in a 6-sided desert plant cawsers are 5n + 1 and 6n, separately. On the off chance that each hexagonal prickly plant has all things considered twofold expurgated vertices and apiece expurgated vertex is pooled by precisely twofold hexagons, at point that desert flora is called hexagonal desert flora hawsers. Twofold vertices are nearby in C6 at that juncture they are on ortho-station, on the off chance that two vertices have separation two, at that point they are on meta-position, and in the event that the separation flanked by twofold vertices is three, at that point that are on para-location. An inner 6-sided in a chain 6-sided prickly plant is baptized ortho-6-sdied, meta-6-sdied, or para-6-sdied if its bowdlerized -vertices are in ortho, meta, and para- location, separately. A normal hawser partakes every single inside hexagon of a hexagonal desert plant chains are of a similar kind.

Theorem 4

For n ≥ 3, the F-index of hexagonal chains are F(On) = F(Mn) = F(Rn) = 96n − 48.

Theorem 5

For n ≥ 3, the square (ortho, para) cactus chains has the minimum F-index than the hexagonal (ortho, meta, para) cactus chains.

Results on polyomino chains

A polyomino framework is a 2-associated chart thru the culmination goalmouth that every single inside aspect is encircled by C4 of length one. The birthplace of polyomino framework goes to the work done by Klarner [21]. The historical backdrop of polyomino framework is ridiculous and extensive that is begun from 20-th century yet they stayed promoted in the current time at first by Golomb [17]. Presently, they are branded by arithmeticians, scientists, physicists and partake been utilized in numerous solicitations [2]. A polyomino fetter is a polyomino context, in which the linking of the focuses of its neighboring standard structures a way c1c2 ···cn, where ci is the focal point of the ith tetragonal [31].

Let Bn be the arrangement of polyomino handcuffs thru stretch n. A C4 of a polyomino chain has it is possible that a couple of neighboring squares. On the off chance that a tetragonal makes them neighbor tetragonal, it is christened lethal, and on the off chance that it has twofold adjacent quadrangles to such an extent that it devours a apogee of gradation 2, it is christened wrinkle. Each restraint of polyomino manacles has 2(n + 1) apogees and 3n + 1 edges. Let BnBn, if the subgraph of Bn incited by the vertices with gradation 3 is a diagram with precisely n − 2 quadrangles, at that point it is known as a straight shackle and meant by Ln [31].

Fig. 4

The zig-zag Z6 and Linear chain L6.

Fig. 5

Segments of a polyomino chain.

A fragment in a polyomino restraints is a maximal straight chain, additionally including wrinkles as well as incurable quadrangles at its end. The extent l(S), 2 ≤ l(S) ≤ n in lieu of polyomino handcuffs of n ≥ 2 plazas, is the quantity of plazas in the ration. The crisscross fetter Zn in polyomino fetters if and just if each portion has extent 2, or state n − 2 snarls.

A polyomino restraints entails of a disarray of fragments S1,S2,S3,...,Sr, r ≥ 1 and l1,l2,l3,...,lr be situated the disarray of extents of fragments, somewhere the entirety of these extents is identical to n + r − 1.

First to compute the F-index, we delineate a stricture α(Si), 1 < i < r, as follows: α(Si)={1forl(Si)=2,0forl(Si)>2, \alpha ({S_i}) = \left\{ {\matrix{ 1 \,\,{{\rm for}\;l({S_i}) = 2{\kern 1pt} ,} \cr 0\,\, { {\rm for}\;l({S_i}) > 2{\kern 1pt} ,} \hfill \cr} } \right. and α(S1) = α(Sr) = 0.

Theorem 6

Let n ≥ 2 be an integer, BnBn be a polyomino restraint per n tetragons and entailing of r fragments signified by S1,S2,...,Sr, (r ≥ 1) with lengths l1,l2,...,lr. Then F(Bn) = 54n + 18r − 40.

Proof

The superiority customary of Bn is divided into twofold detachments: E1 and E2. The detachments E1 comprehends, every such edge which are expurgated athwart by conservative ran line going through the focuses of Si for 1 ≤ Ir (Figure 6(a)). The detachments E2 comprises of such edges that are not constituent of E1, i.e., E2 = E(Bn) \ E1, the components of E2 are portrayed in Figure 6(b) as a conventional run streaks. At that point, we get uvE1[(Θ(u))2+(Θ(v))2]=i=1ruvE1E(Si)[(Θ(u))2+(Θ(v))2]. \sum\limits_{uv \in {E_1}} [(\Theta (u{))^2} + {(\Theta (v))^2}] = \sum\limits_{i = 1}^r \sum\limits_{uv \in {E_1} \cap E({S_i})} [(\Theta (u{))^2} + {(\Theta (v))^2}]. For 1 < i < r, we have: uvE1E(Si)[(Θ(u))2+(Θ(v))2]=18(li2)+7α(Si1)+7α(Si1)+58,uvE1E(S1)[(Θ(u))2+(Θ(v))2]=18(l12)+7α(S2)+46,uvE1E(Sr)[(Θ(u))2+(Θ(v))2]=18(lr2)+7α(Sr1)+46. \matrix{ {\sum\limits_{uv \in {E_1} \cap E({S_i})} [(\Theta (u{{))}^2} + {{(\Theta (v))}^2}]} \,\, { = 18({l_i} - 2) + 7\alpha ({S_{i - 1}}) + 7\alpha ({S_{i - 1}}) + 58,} \hfill \cr {\sum\limits_{uv \in {E_1} \cap E({S_1})} [(\Theta (u{{))}^2} + {{(\Theta (v))}^2}]} \,\, { = 18({l_1} - 2) + 7\alpha ({S_2}) + 46,} \hfill \cr {\sum\limits_{uv \in {E_1} \cap E({S_r})} [(\Theta (u{{))}^2} + {{(\Theta (v))}^2}]} \,\, { = 18({l_r} - 2) + 7\alpha ({S_{r - 1}}) + 46.} \hfill \cr } Therefore, we have uvE1[(Θ(u))2+(Θ(v))2]=18i=1r(li2)+14i=2r1α(Si)+58r24. \sum\limits_{uv \in {E_1}} [(\Theta (u{))^2} + {(\Theta (v))^2}] = 18\sum\limits_{i = 1}^r ({l_i} - 2) + 14\sum\limits_{i = 2}^{r - 1} \alpha ({S_i}) + 58r - 24. Also we have: uvE2[(Θ(u))2+(Θ(v))2]=i=1ruvE2E(Si)(Θ(u)2+Θ(v)2). \sum\limits_{uv \in {E_2}} [(\Theta (u{))^2} + {(\Theta (v))^2}] = \sum\limits_{i = 1}^r \sum\limits_{uv \in {E_2} \cap E({S_i})} (\Theta {(u)^2} + \Theta {(v)^2}). For 1 < i < r, we have uvE2E(Si)[(Θ(u))2+(Θ(v))2]=18(2li4)14α(Si)+14,uvE2E(S1)[(Θ(u))2+(Θ(v))2]=18(2l14)+33,uvE2E(Sr)[(Θ(u))2+(Θ(v))2]=18(2lr4)+33. \matrix{ {\sum\limits_{uv \in {E_2} \cap E({S_i})} [(\Theta (u{{))}^2} + {{(\Theta (v))}^2}]} \,\, { = 18(2{l_i} - 4) - 14\alpha ({S_i}) + 14,} \hfill \cr {\sum\limits_{uv \in {E_2} \cap E({S_1})} [(\Theta (u{{))}^2} + {{(\Theta (v))}^2}]} \,\, { = 18(2{l_1} - 4) + 33,} \hfill \cr {\sum\limits_{uv \in {E_2} \cap E({S_r})} [(\Theta (u{{))}^2} + {{(\Theta (v))}^2}]} \,\, { = 18(2{l_r} - 4) + 33.} \hfill \cr } Therefore, we get uvE2[(Θ(u))2+(Θ(v))2]=18i=1r(2li4)14i=2r1α(Si)+14r+38. \sum\limits_{uv \in {E_2}} [(\Theta (u{))^2} + {(\Theta (v))^2}] = 18\sum\limits_{i = 1}^r (2{l_i} - 4) - 14\sum\limits_{i = 2}^{r - 1} \alpha ({S_i}) + 14r + 38. F(Bn)=uvE(Bn)[(Θ(u))2+(Θ(v))2]=uvE1[(Θ(u))2+(Θ(v))2])+uvE2[(Θ(u))2+(Θ(v))2]=54n+18r40. \matrix{ {F({B_n})} \hfill & { = \sum\limits_{uv \in E({B_n})} [(\Theta (u{{))}^2} + {{(\Theta (v))}^2}]} \hfill \cr {} \hfill & { = \sum\limits_{uv \in {E_1}} [(\Theta (u{{))}^2} + {{(\Theta (v))}^2}]) + \sum\limits_{uv \in {E_2}} [(\Theta (u{{))}^2} + {{(\Theta (v))}^2}]} \hfill \cr {} \hfill & { = 54n + 18r - 40.} \hfill \cr }

Fig. 6

(a) The edges of E1. (b) The edges of E2.

Fig. 7

Z5 {Z_5^\prime} and A6.

Theorem 7

For n ≥ 2, then the F-index of linear and zig-zag chains are

F(Ln) = 54n − 22.

F(Zn) = 72n − 58.

Theorem 8

For n ≥ 2, then F(Ln) ≤ F(Bn) ≤ F(Zn), with right(left) equality if and only if BnZn (LnBn).

Proof

From the Theorems 6 and 7, it can be easily seen that Bn is thoroughgoing(minutest) if and only if r is thoroughgoing(minutest). Clearly, we have BnZn (LnBn) if and only if r is thoroughgoing(minutest).

We presently figure and afterward look at the F-record of crisscross and easy chair nanoribbons with n hexagons. The charts of crisscross and easy chair nanoribbons are meant by Zn {Z_n^\prime} and An, individually. The formulae of the F-list of these charts are the followings:

Theorem 9

For n ≥ 2, we have

F(Zn)=70n22 F({Z_n^\prime}) = 70n - 22 .

F(An) = 70n + 58.

From Theorem 9, we get the minimum F-index of nanoribbons which is stated in the following theorem.

Theorem 10

For n ≥ 2, we have F(Zn)<F(An) F({Z_n^\prime}) < F({A_n}) .

Conclusion

In this paper, we partake figured the F-index of triangular cactus manacles, tetragonal cactus restraints, hexagonal cactus restraints and the polyomino restraints. We correspondingly strongminded the tiniest cactus restraints by means of reverence to the F-index. Nevertheless, we strongminded the extremal polyomino restraints per veneration to F-index.

Fig. 1

The graphs T5 and W4.
The graphs T5 and W4.

Fig. 2

The ortho-chain S8 and para-chain Q6.
The ortho-chain S8 and para-chain Q6.

Fig. 3

The ortho-chain, meta-chain and para chain.
The ortho-chain, meta-chain and para chain.

Fig. 4

The zig-zag Z6 and Linear chain L6.
The zig-zag Z6 and Linear chain L6.

Fig. 5

Segments of a polyomino chain.
Segments of a polyomino chain.

Fig. 6

(a) The edges of E1. (b) The edges of E2.
(a) The edges of E1. (b) The edges of E2.

Fig. 7

Z5′
{Z_5^\prime}


 and A6.
Z5′ {Z_5^\prime} and A6.

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