The

#### Keywords

- Forgotten index
- Zagreb index
- cactus chain
- polyomino chain

#### MSC 2010

- 57M15

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

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

In the event that

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

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 _{n}_{n}_{5} is appeared in Figure 1. Next, we process the

_{n}

The windmill diagram signified by _{n}_{n}_{n}

_{n}^{3} + 16

A Square Cactus Chains is a diagram with tetragons are _{4} i.e., 4-undeviating. The quantity of vertices and the quantity of edges of the square prickly plant are 3_{4}, at that point such tetragonal is an ortho- tetragons and is meant by _{n}_{4}, at that point such tetragons are christened para-tetragons and is indicated by _{n}

_{n}_{n}

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 5_{6} 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.

_{n}_{n}_{n}

A polyomino framework is a 2-associated chart thru the culmination goalmouth that every single inside aspect is encircled by _{4} 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 _{1}_{2} _{n}_{i}

Let _{n}_{4} 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}_{n}_{n}_{n}

A fragment in a polyomino restraints is a maximal straight chain, additionally including wrinkles as well as incurable quadrangles at its end. The extent _{n}

A polyomino restraints entails of a disarray of fragments _{1},_{2},_{3},...,_{r}_{1},_{2},_{3},...,_{r}

First to compute the _{i}_{1}) = _{r}

_{n}_{n}_{1},_{2},...,_{r},_{1},_{2},...,_{r}. Then F_{n}

The superiority customary of _{n}_{1} and _{2}. The detachments _{1} comprehends, every such edge which are expurgated athwart by conservative ran line going through the focuses of _{i}_{2} comprises of such edges that are not constituent of _{1}, i.e., _{2} = _{n}_{1}, the components of _{2} are portrayed in Figure 6(b) as a conventional run streaks. At that point, we get

_{n}

_{n}

_{n}_{n}_{n}_{n}_{n} (L_{n}_{n}).

From the Theorems 6 and 7, it can be easily seen that _{n}_{n}_{n}_{n}_{n}

We presently figure and afterward look at the _{n}

_{n}

From Theorem 9, we get the minimum

In this paper, we partake figured the

_{1}-index and its applications to fullerenes. _{1}-index and its applications to fullerenes

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