On detour index of cycloparaphenylene and polyphenylene molecular structures

Cycloparaphenylene is a particle that comprises a few benzene rings associated with covalent bonds in the para positions to frame a ring-like structure. Similarly, poly (para-phenylenes) are macromolecules that include benzenoid compounds straightforwardly joined to each other by C–C bonds. Because of their remarkable architectural highlights, these structures have fascinated attention from numerous vantage focuses. Descriptors are among the most fundamental segments of prescient quantitative structure-activity and property relationship (QSAR/QSPR) demonstrating examination. They encode chemical data of particles as quantitative numbers, which are utilized to create a mathematical correlation. The nature of a predictive model relies upon great demonstrating insights, yet additionally on the extraction of compound highlights. To a great extent, Molecular topology has exhibited its adequacy in portraying sub-atomic structures and anticipating their properties. It follows a two-dimensional methodology, just thinking about the interior plan, including molecules. Explicit subsets speak the design of every atom of topological descriptors. When all around picked, these descriptors give a unique method of describing an atomic system that can represent the most significant highlights of the molecular structure. Detour index is one such topological descriptor with much application in chemistry, especially in QSAR/QSPR studies. This article presents an exact analytical expression for the detour index of cycloparaphenylene and poly (para-phenylene).

There are different kinds of carbon-based nanostructures such as carbon nanorings 18,19 , nanosprings 20 , and nanocones 21 , etc. Carbon nanorings have been seen in single-walled carbon nanotubes (SWCNTs) developed by laser vaporization 22 . The diameters of these round structures range amid 300 and 500 nm and their widths somewhere in the range of 5 and 15 nm. There are no topological pentagon-heptagon deserts in these structures, as kinks that could be made by such imperfections are not watched. In this manner, they can be imaged as the bowing of a straight SWCNT into a ring by associating its two closures to shape the carbon nanorings. These nanoring structures might be ideal nanodevices because of their interesting mechanical 13 , and physical properties 19 . Among the most limited formed piece of armchair carbon nanotubes (CNTs), cycloparaphenylenes (CPPs) have as of late pulled in expanding consideration from scientists. CPPs have straightforward hoop-shaped structures comprising aromatic rings with para-linkage, which were guessed 50 years ago, yet blended distinctly in the last decade 23 . Their stressed and contorted aromatic frameworks and radially arranged p orbitals have fascinated manufactured physicists, theoreticians, supramolecular scientific experts, and materials researchers the same. In spite of this boundless importance, the CPPs remain an difficult synthetic challenge. It is trying to make brilliant, stable CPPs with a little HOMO-LUMO gap because of restricting strain based reactivity and symmetry-based fluorescence extinguishing for little CPPs 24 . A few exploration bunches have created combinations of [n] CPPs of distinctive ring sizes (here n speaks to the quantity of benzene) as depicted in Fig. 1. Different techniques have incredibly researched the impact of [n]CPPs on the microelectronic stuff 25 . The sum of characteristic polynomial of [n] CPPs were reported in 26 .
Linear poly(para-phenylene) (PPP) are polymeric compounds with hexagonal rings as reproducing units as shown in Fig. 2 and are significant polycyclic aromatic compounds that are discovered to be the fundamental units of numerous novel materials like graphene or related compounds. Because of these, linear PPPs have been the focal point of fascination for both experimentalists and chemists [27][28][29] since the most recent couple of many years. Linear PPPs and their derivatives have broadly been utilized in optoelectronic applications. Although there are a variety of linear polyaromatic polymers 30 , linear polyphenylenes are extremely insoluble unless they have solubilizing functional groups that can form hydrogen bonds with water. For instance, functional groups such as OH, NH 2 , and COOH.
Topological indices (TIs) of huge chemical structures, for example, metal natural systems can be amazingly valuable in both portrayal of structures and processing their physicochemical properties that are generally difficult to calculate for such enormous organizations of significance in reticular chemistry. It is a mathematical quantity which bonds molecular topology to molecular properties 31 . Such entities are invariants of graph and are utilized as descriptors for QSAR/ QSPR examinations 32,33 , proven to be an vigorous zone at the frontline of research. These descriptors are exceptionally valuable for looking through database of molecule, predicting molecular properties 34 , screening of drug 5 , designing of drug 9 , complex networks 35,36 and many other procedures. The idea of this molecular descriptor came from Wiener's effort in 1947 37 . He detected that there is a high degree of correlation amid the melting point with the Wiener index 38-40 .

Background
Graph theory is a field of mathematics with potential application in engineering and science 41 . The theory provides a solid foundation for investigating topological requirements of many systems. Kaveh and Koohestani 42,43 have effectively applied graph theory to the optimal analysis of FEMs in the framework of the force method in structural mechanics. Graph theory's practical and beneficial applications include visualisation of sparse matrices, nodal ordering, envelope reduction, graph partitioning, and configuration processing. The reader who is interested can search up where the   www.nature.com/scientificreports/ majority of these applications have been recorded in 44,45 . We use the theory to generate reducible representations of symmetry groups, taking into account the unique specifications of graphs. Let |V(G)| and |E(G)| be the number of vertices and edges of a chemical graph G respectively. For any two vertices x and y are adjacent if there is an edge between them. Distance between two vertices x, y ∈ V (G) is the number of edges in the shortest path connecting them in a connected graph G and is denoted by d G (x, y) 46,47 . Similarly, the detour distance 48,49 among two vertices x, y ∈ V (G) is the number of edges in the longest path connecting them in a connected graph G and is denoted by l G (x, y) . Also with the note d G (u, u) = 0 and l G (u, u) = 0 , the transmission (farness or vertex Wiener value) of a vertex u ∈ V (G) defined by W(u), as the sum of the lengths of all shortest paths between u to all other vertices in G [50][51][52][53] . Following this, we define the detour transmission (vertex detour value) of a vertex u ∈ V (G) is denoted by ω(u) and explained as the sum of the detour lengths of all longest paths between u to all other vertices in G. Mathematically, and The Wiener index W(G) is the sum of shortest distance between every pair of vertices, where as the detour index ω(G) is the sum of longest distance between every pair of vertices 4854, 55 . Mathematically, and The application of detour index in QSAR considers is clarified by Lukovits in 56 . Rücker 57 additionally researched this idea as a invariant for melting points of alkanes of cyclic and acyclic nature. It is noticed that Wiener index and detour index are equivalent if and only if G is acyclic and there are a few research papers on Wiener index of trees with a given condition and those result hold for detour index. It merits researching the detour index of cyclic graphs. For additional details on this investigation refer [58][59][60][61] .
In 60 the authors derived an algorithm for recognizing the longest path among any pair of vertices of a graph and it was utilized to calculate an exact analytical formula for the detour index of fused bicyclic networks. Computer strategies for computing the detour distances and subsequently for calculating the detour index was derived in 54,57 . It has been demonstrated in 62 , and the detour matrix is a NP-complete problem. A strategy for building the detour matrix 63,64 for graphs of modest sizes were introduced in 65 . Inter correlation amid hyper-detour index and other TIs such as Wiener, Harary, hyper-Wiener, hyper-Harary, and detour index were evaluated in 60 on three pairs of branched and unbranched. Cycloalkanes and alkanes and with up to eight carbon particles and the hyper-detour index have been examined in structure-property studies 33 . Ongoing applications of the hyperdetour index discovered in 66 .
The detour index has also had great success combined with the Wiener index in structure boiling point modelling of cyclic and acyclic hydrocarbons. In 54 the authors analyzed the importance of the detour index and correlated its application with the Wiener index. Also, they established that the detour index combined with the Wiener index is very adequate in the structure-boiling point modelling of acyclic and cyclic saturated hydrocarbons. This achievement has prompted the advancement of related indices such as the hyper-detour index 67 and the Cluj-detour index 59 . Qi and Zhou 67 presented the hyper-detour index of unicyclic graphs and decided the graphs with the smallest and biggest detour indices respectively in the class of n-vertex bicyclic graphs with precisely two cycles for n ≥ 5 . In 68 , Du decided the graphs with the second and the third smallest and biggest detour indices in the class of n-vertex bicyclic graphs with precisely two cycles for n ≥ 6 . Very recently Prabhu et al. have found the detour index for join of graph 69 . This paper presents an expression for the detour index of cycloparaphenylene and poly (p-phenylene) using detour transmission of a vertex.

Results
In Ref. 61 , the experimental and calculated boiling points (°C) of 76 alkanes and cycloalkanes, as well as their Wiener and Detour indices, are reported. For acyclic structures, the Wiener index W and the detour index ω are, of course, identical. W and ω are not very intercorrelated indices for polycyclic structures. The linear correlation between W and ω (ω = aW + b) for a set of 37 diverse polycyclic graphs was presented with a modest correlation coefficient (r = 0.79) in Ref. 55 , while the exponential relationship between W and ω (ω = aW b ) produced only a slightly better correlation between them (r = 0.86). With this motivation, we proceed to find the detour index of the CPP and PPP. In this section, we first explain the vertex set and edge set of cycloparaphenylene and poly (p-phenylene) before proceeding to our main results.
It is observed from the molecular structure of cycloparaphenylene [n]−CPP and polyphenylene PPP(n) the vertex set of these two molecular graphs remains same and is given by www.nature.com/scientificreports/ with |j − i| = 1 or n − 1 and for PPP(n), it is And their cardinalities are respectively given by 7n and 7n − 1 . The molecular graph of cycloparaphenylene and polyphenylene were depicted in Fig. 3a,b. Fig. 4a. Clearly Fig. 4b depicts the path of length 4(n − i + 1) for i ≤ ⌈ n 2 ⌉ .   www.nature.com/scientificreports/ Fig. 8.

Lemma 2 Let G be a molecular graph of a cycloparaphenylene and {a
a 1 a n ' c n ' b n ' a n a n-1 ' a i+1 a i ' a 1 a n ' c n ' b n ' a n a n-1 ' a i+1 a i ' a 1 a n ' c n ' b n ' a n a n-1 ' a i+1 a i '  www.nature.com/scientificreports/ Fig. 9.
For details refer Fig. 10.
The Hamilton-path construction is depicted in Fig. 11.
The following lemma is straight forward from the structural property of [n]-CPP and the addressing scheme proposed in the begening of this section.    www.nature.com/scientificreports/ Proof For n even, the detour transmission of a 1 ∈ V (G) is given by,

Lemma 4 Let G be a molecular graph of a cycloparaphenylene and {a
(4i + 1) www.nature.com/scientificreports/ For n even, the detour transmission of b 1 ∈ V (G) For n is odd, lG(b1, ai) + n i= n+3 2 lG(b1, ai) with the similar argument along with Lemma 4, we derive the result for n odd.

Lemma 5 Let G be a molecular graph of a linear polyphenylene of dimension n, and {a
Proof For any n and a i ∈ V (G) , the detour transmission of a i is given by ω(G) = 54n 3 + 60n 2 − 72n if n is even 54n 3 + 60n 2 − 71n if n is odd www.nature.com/scientificreports/ For b i ∈ V (G) , the detour transmission of b i is given by And for c i ∈ V (G) , the detour transmission of c i is given by www.nature.com/scientificreports/ Now for a ′ i ∈ V (G) , the detour transmission of a vertex a ′ i is Theorem 2 Let G be a molecular graph of linear polyphenylene of dimension n. Then ω(G) = 24n 3 + 72n 2 − 33.
Proof Due to symmetry, for any b i , The graphical representation of the detour index of cycloparaphenylene CPP(n) and poly (p-phenylene) PPP(n) were depicted in Fig. 12, which says that the detour index of cycloparaphenylene is higher than polyphenylene irrespective of n.

Conclusion
In recent decade, CPPs have gone from being manufactured interests to promptly open materials with exceptionally tunable properties. The syntheses of CPPs are motivated by a wide extent of energizing applications, going from strong state nanomaterials to organic imaging. Also, the aromatic polymers of PPPs comprising of straightforwardly repeating benzene units as their spine. PPP has interesting optical properties, for example, electroluminescence, and is regularly utilized as tunable blue-transmitting material for light-radiating devices. Detour index is a promising topological index and the study of this index is very helpful to acquire the basic topologies of networks. We accept that the detour index acquired here well correspond with a portion of the physico-chemical properties and a portion of the structure-property relations.