Computing Forgotten Topological Index of Extremal Cactus Chains

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)=∑v∈V(Θ(v))2=∑uv∈E(Θ(u)+Θ(v)),M2(G)=∑uv∈E[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)=∑v∈V(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)=∑v∈VΘ(v)α=∑uv∈E[(Θ(u))(α−1)+(Θ(v))(α−1)]. 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)=∑u∈V(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].

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].

2.1

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 T_n, 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 T_n 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

Theorem 1

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

The windmill diagram signified by W_n 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 W_n 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 W_n.

Theorem 2

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

2.2

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 S_n. In the event that twofold pinnacles are not immediate in C₄, at that point such tetragons are christened para-tetragons and is indicated by Q_n (These phrasings engaged after the hypothesis of benzenoid hydrocarbons [11]. The ortho and para restraints of tetragons are appeared in Figure 2.

Fig. 2

Theorem 3

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

Fig. 3

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 ···c_n, where c_i is the focal point of the ith tetragonal [31].

Let B_n 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 B_n ∈ B_n, if the subgraph of B_n 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 L_n [31].

Fig. 4

Fig. 5

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 Z_n 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,...,S_r, r ≥ 1 and l₁,l₂,l₃,...,l_r 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(S_i), 1 < i < r, as follows: α(Si)={1 for l(Si)=2,0 for l(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(S_r) = 0.

Fig. 6

Fig. 7

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 A_n, individually. The formulae of the F-list of these charts are the followings:

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

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.