On the Irregularity of Some Molecular Structures

TL;DR

This paper analyzes various irregularity measures of chemical and molecular graphs, including nanotubes and dendrimers, to understand their properties and potential applications in QSAR/QSPR studies.

Contribution

It provides a detailed structural analysis and derivation of irregularity measures for a range of chemical molecular graphs and nanotubes, expanding understanding of their properties.

Findings

Irregularity measures vary across different molecular structures.

Structural analysis reveals patterns correlating irregularity with chemical properties.

Mycielski's constructions exhibit distinct irregularity characteristics.

Abstract

Measures of the irregularity of chemical graphs could be helpful for QSAR/QSPR studies and for the descriptive purposes of biological and chemical properties, such as melting and boiling points, toxicity and resistance. Here we consider the following four established irregularity measures: the irregularity index by Albertson, the total irregularity, the variance of vertex degrees and the Collatz-Sinogowitz index. Through the means of graph structural analysis and derivation, we study the above-mentioned irregularity measures of several chemical molecular graphs which frequently appear in chemical, medical and material engineering, as well as the nanotubes: $TUC_4 C_8(S)$, $TUC_4 C_8(R)$, Zig-Zag $TUHC_{6}$, $TUC_4$, Armchair $TUVC_{6}$, then dendrimers $T_{k,d}$ and the circumcoronene series of benzenoid $H_k$. In addition, the irregularities of Mycielski's constructions of cycle and path graphs are analyzed.