Numerical and Scientific Investigation of Some Molecular Structures Based on the Criterion of Super Classical Average Assignments

Abstract

Numbering a graph is a very practical and effective technique in science and engineering. Numerous graph assignment techniques, including distance-based labeling, topological indices, and spectral graph theory, can be used to investigate molecule structures. Consider the graph $G$, with the injection $\Omega$ from the node set to $\left\{1,2,\dots ,∆\right\}$, where $∆$ is the sum of the number of nodes and links. Assume that the induced link assignment ${\Omega }^{\ast }$ is the ceiling function of the average of root square, harmonic, geometric, and arithmetic means of the vertex labels of the end vertices of each edge. If the union of range of $\Omega$ of the node set and the range of $\Omega$ of the link set is the set $\left\{1,2,\dots ,∆\right\}$, then $\Omega$ is called a super classical average assignment (SCAA). This is known as the SCAA criterion. In this study, the graphical structures corresponding to chemical structures based on the SCAA criterion are demonstrated. The graphical depiction of chemical substances was first defined and second, the union of any number of cycles $C_n$, the tadpole graph, the graph extracted by identifying a line of any two cycles $C_m$ and $C_n$, and the graph extracted by joining any two cycles by a path are all examined in this work.