AUTHOR=Luo Ricai , Khalil Adnan , Ahmad Ali , Azeem Muhammad , Ibragimov Gafurjan , Nadeem Muhammad Faisal 

TITLE=Computing the partition dimension of certain families of Toeplitz graph

JOURNAL=Frontiers in Computational Neuroscience

VOLUME=16

YEAR=2022

URL=https://www.frontiersin.org/journals/computational-neuroscience/articles/10.3389/fncom.2022.959105

DOI=10.3389/fncom.2022.959105

ISSN=1662-5188

ABSTRACT=<p>Let <italic>G</italic> = (<italic>V</italic>(<italic>G</italic>), <italic>E</italic>(<italic>G</italic>)) be a graph with no loops, numerous edges, and only one component, which is made up of the vertex set <italic>V</italic>(<italic>G</italic>) and the edge set <italic>E</italic>(<italic>G</italic>). The distance <italic>d</italic>(<italic>u, v</italic>) between two vertices <italic>u, v</italic> that belong to the vertex set of <italic>H</italic> is the shortest path between them. A <italic>k</italic>-ordered partition of vertices is defined as β = {β<sub>1</sub>, β<sub>2</sub>, …, β<sub><italic>k</italic></sub>}. If all distances <italic>d</italic>(<italic>v</italic>, β<sub><italic>k</italic></sub>) are finite for all vertices <italic>v</italic> ∈ <italic>V</italic>, then the <italic>k</italic>-tuple (<italic>d</italic>(<italic>v</italic>, β<sub>1</sub>), <italic>d</italic>(<italic>v</italic>, β<sub>2</sub>), …, <italic>d</italic>(<italic>v</italic>, β<sub><italic>k</italic></sub>)) represents vertex <italic>v</italic> in terms of β, and is represented by <italic>r</italic>(<italic>v</italic>|β). If every vertex has a different presentation, the <italic>k</italic>-partition β is a resolving partition. The partition dimension of G, indicated by <italic>pd</italic>(<italic>G</italic>), is the minimal <italic>k</italic> for which there is a resolving <italic>k</italic>-partition of <italic>V</italic>(<italic>G</italic>). The partition dimension of Toeplitz graphs formed by two and three generators is constant, as shown in the following paper. The resolving set allows obtaining a unique representation for computer structures. In particular, they are used in pharmaceutical research for discovering patterns common to a variety of drugs. The above definitions are based on the hypothesis of chemical graph theory and it is a customary depiction of chemical compounds in form of graph structures, where the node and edge represent the atom and bond types, respectively.</p>