Study on (r,s)- Generalised Transformation Graphs, A Novel Perspective Based on Transformation Graphs

TL;DR

This paper introduces a new class of graphs called (r,s)-generalised transformation graphs, extending existing transformation graphs to analyze structural features of graphs and compounds using the first Zagreb index.

Contribution

The work extends transformation graphs to (r,s)-generalised transformation graphs, allowing analysis of infinite graph classes based on vertex and edge copies.

Findings

Defined (r,s)-generalised transformation graphs.

Analyzed these graphs using the first Zagreb index.

Established the potential for modeling complex structures.

Abstract

For a graph $\mathbb{Q}=(\mathbb{V},\mathbb{E})$, the transformation graphs are defined as graphs with vertex set being $\mathbb{V(Q)} \cup \mathbb{E(Q)}$ and edge set is described following certain conditions. In comparison to the structure descriptor of the original graph $\mathbb{Q}$, the topological descriptor of its transformation graphs displays distinct characteristics related to structure. Thus, a compound's transformation graphs descriptors can be used to model a variety of structural features of the underlying chemical. In this work, the concept of transformation graphs are extended giving rise to novel class of graphs, the $(r,s)$- generalised transformation graphs, whose vertex set is union of $r$ copies of $\mathbb{V(Q)}$ and $s$ copies of $\mathbb{E(Q)}$, where, $r, s \in N$ and the edge set are defined under certain conditions. Further, these class of graphs are analysed with the help of first Zagreb index. Mainly, there are eight transformation graphs based on the criteria for edge set, but under the concept of $(r,s)$- generalised transformation graphs, infinite number of graphs can be described and analysed.