Irregularity Measures for Benzene Ring Embedded in P-Type Surface

School of Electronic Engineering, Huainan Normal University, Huainan 232038, China Institute of Chemistry, University of the Punjab, Lahore, Pakistan Department of Mathematics, Huzhou University, Huzhou 313000, China Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, China Department of Mathematics, University of Management and Technology, Lahore, Pakistan


Introductioń
OKeeffe et al. [1] have dispersed around a quarter century ago a letter executing two 3D classification of benzene. From which, one is called 6.82P (polybenzene) and has a place with the space gather Im3m, contrast to the P-type surface, and this is due to an insertion of the hexagon fix in the surface of negative ebb and flow P. e P-type surface is facilitated to the Cartesian organizes in the Euclidean space. For further detail about this recurring surface, the author is referred to [2,3]. is structure needed to be joined as 3D carbon solids; be that as it may, according to our knowledge, no such sequence was assumed before. e goal was to provoke the devotion of scientists to the atomic acknowledgment of such amiable thoughts in carbon nanoscience, as the graphenes took up a moment Nobel prize after C60, and also the immediate union of fullerenes is presently a reality, see for detail [4,5].
Graph theory provides an interesting appliance in mathematical chemistry where it is used to compute the various kinds of chemical compounds and predict their various properties. One of the most important tools in the chemical graph theory is the topological index, which is useful in predicting the chemical and physical properties of the underlying chemical compound, such as boiling point, strain energy, rigidity, heat of evaporation, and tension [6,7]. A graph having no loop or multiple edge in known as a simple graph. A molecular graph is a simple graph in which atoms and bounds are represented by vertices and edges, respectively. e degree of the vertex is the number of edges attached with that vertex. ese properties of various objects are of primary interest. Winner, in 1947, introduced the concept of the first topological index while finding the boiling point. In 1975, Gutman gave a remarkable identity [8] about Zagreb indices. Hence, these two indices are among the oldest degree-based descriptors, and their properties are extensively investigated. e mathematical formulae of these indices are (1) A topological index is known as an irregularity index [9] if the value of the topological index of the graph is greater than or equal to zero, and the topological index of the graph is equal to zero if and only if the graph is regular. e irregularity indices are given in Table 1. Most of the irregularity indices are from the family of degree-based topological indices and are used in quantitative structure activity relationship modeling.
For more about topological indices, one can read [10-33].

Irregularity Indices for BR p
is section is about irregularity indices of BR p . e molecular graph of BR p is given in Figure 1. We can observe from Figure 1 that there are two types of vertices present in the molecular graph of BR p i.e., 2 and 3. e cardinality of the edge set is 32pq − 2p − 2q. e edge partition of BR p is given in Table 2.
Mathematical Problems in Engineering 3

Theorem 2. Let G be SBR p . e irregularity indices are
toxicity, resistance, and entropy. It is hoped that this article will aid the reader in understanding the rationale and utility of a simple quantitative tool which could be used in malocclusion assessment.

Data Availability
All data are including in this paper.

Conflicts of Interest
e authors do not have conflicts of interest.