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Abstract Nanosheets with boron elements have excellent characteristics which makes the boron polymorphs unique and super hard. A boron $$\alpha$$ α -icosahedral nanosheet in crystalline form has superconductivity and thermal electronic properties. In theoretical chemistry and QSPR/QSAR study, a topological descriptor is an important analytical tool. It helps to analyse the structure and its properties and also correlates the with numerical expressions. The valence-based M-polynomial provides quantitative measures of molecular properties based on their geometric, electrostatic, and quantum chemical characteristics. In this article, the QSPR/QSAR analysis is performed for this nanosheet and the analytical expressions are validated with original synthesized data, and received excellent correlation values of 0.9835 and 0.9932. The mathematical expression of the structure is analysed and the indices are compared graphically and numerically.

For a graph X=(V(X),E(X)), a wide range of transformations are possible. A graph X can be converted into a new graph by adding vertices or edges following certain criteria. A graph derived from the parent graph X by inserting a new vertex into each of its edge is known as subdivision graph and can be denoted as S(X). The transformation graph of a graph X can be defined as a graph whose vertex set is union of the vertex set and edge set of X and whose edge set is defined under certain conditions. Structure descriptors are widely used now-a-days to model the physico-chemical, structural and electronic properties of chemical compounds and networks. A structure descriptor of a transformation graph reflects different structural features than a structure descriptor of the original graph X. By determining various structure descriptors of the transformation graphs of a molecule, wide range of structural properties of the underlying molecule could be modelled. In this study, the concept of transformation graph is applied on subdivision graph and further the structural properties of those graphs are studied via structure descriptors. The expressions for first Zagreb index and forgotten index of transformation graphs of subdivision graphs are determined in terms of parameters of the parental graph. Further, applying the determined results, a structural analysis was also carried out on a chemical graph.

Abstract Nanomaterials feature exceptional, one-of-a-kind qualities that might be used in electronics, medicine, and other industries. Two-dimensional nanomaterials called borophene have a variety of intriguing characteristics, which helped them to leave an indelible impression in the fields of chemistry, material science, nanotechnology, and condensed matter physics. The concept of modelling the structure of a molecule or chemical network to a chemical graph and then quantitatively analysing them with the aid of topological descriptors was a major advance in the fields of mathematics and chemistry, with a wide range of applications. M-polynomial approach is a very versatile and quick method for computing the degree-based descriptors of chemical graphs or networks. The degree-based descriptors of the $$\beta _{12}$$ β 12 -Borophene nanosheet are established in this study utilising the M-polynomial technique. A program code that enables to generate the M-polynomial of any chemical structure was developed in Java platform and the same is displayed. At the conclusion, the numerical and graphical comparison based on the identified analytic expressions is also provided. Additionally, the QSPR analysis was also carried out and the outcoms are presented therein.

In this article, a novel technique to evaluate and compare the neighborhood degree molecular descriptors of two variations of the carbon nanosheet C5C7(a,b) is presented. The conjugated molecules follow the graph spectral theory, in terms of bonding, non-bonding and antibonding Ruckel molecular orbitals. They are demonstrated to be immediately determinable from their topological characteristics. The effort of chemical and pharmaceutical researchers is significantly increased by the need to conduct numerous chemical experiments to ascertain the chemical characteristics of such a wide variety of novel chemicals. In order to generate novel cellular imaging techniques and to accomplish the regulation of certain cellular mechanisms, scientists have utilized the attributes of nanosheets such as their flexibility and simplicity of modification, out of which carbon nanosheets stand out for their remarkable strength, chemical stability, and electrical conductivity. With efficient tools like polynomials and functions that can forecast compound features, mathematical chemistry has a lot to offer. One such approach is the M-polynomial, a fundamental polynomial that can generate a significant number of degree-based topological indices. Among them, the neighborhood M-polynomial is useful in retrieving neighborhood degree sum-based topological indices that can help in carrying out physical, chemical, and biological experiments. This paper formulates the unique M-polynomial approach which is used to derive and compare a variety of neighborhood degree-based molecular descriptors and the corresponding entropy measures of two variations of pent-heptagonal carbon nanosheets. Furthermore, a regression analysis on these descriptors has also been carried out which can further help in the prediction of various properties of the molecule.

Let G be a graph with vertex set as VG and edge set as EG which is simple as well as connected. The problem of strong total monophonic set is to find the set of vertices T⊆VG, which contains no isolated vertices, and all the vertices in VG\T lie on a fixed unique chordless path between the pair of vertices in T. The cardinality of strong total monophonic set which is minimum is defined as strong total monophonic number, denoted by smtG. We proved the NP-completeness of strong total monophonic set for general graphs. The strong total monophonic number of certain graphs and networks is derived. If l,m,n are positive integers with 5≤l≤m≤n and m≤2l−1, then we can construct a connected graph G with strong monophonic number l and strong total monophonic number m.

Topological indices are numeric parameters which portray the topology of a subatomic structure. In QSAR/QSPR analysis, topological descriptors play a vital role to examine the topology of a network. An interconnection network is a structure whose components are connected physically according to some pattern. In this paper, an interconnection network, ternary hypertree, which is a structural combination of complete ternary tree and hypertree, is introduced. We have evaluated the topological descriptors grounded on the distances for the ternary hypertree. The analytical expressions for Wiener, different types of Szeged, and Mostar indices are determined.