Vertex-Edge Partition Resolvability for Certain Carbon Nanocones

Abstract

Chemical graph theory is an extension of mathematical chemistry that explores chemical phenomena and entities using the conceptual frameworks of graph theory. In particular, chemical graphs are used to represent molecular structures in chemical graph theory. Edges and vertices in this chemical graph substitute for bonds and atoms, respectively. The primary data types used throughout cheminformatics to depict chemical structures are chemical graphs. The basis for (quantitative) structure-property and structure-activity predictions, a central area of cheminformatics, is laid by the computable properties of graphs. The physical characteristics of molecules are thus reflected in these graphs, which can subsequently be reduced to graph-theoretical indices or descriptors. The resolving set $Z,$ which distinguish every pair of distinct vertices in a connected simple graph, is one of the most well-studied distance-based graph descriptors. In this manuscript, we consider the most significant variation of a metric dimension known as partition dimension and determine it for the molecular complex graph of one-pentagonal carbon nanocone (PCN). We demonstrate that all of the vertices and edges in PCN can be uniquely identified only by considering three distinct non-neighboring partition sets (minimal requirement) from PCN.

Keywords:

• One-pentagonal carbon nanocone
• basis set
• resolving set
• partition basis set
• partition dimension
• molecular graph

Acknowledgments

The author would like to express their sincere thanks to the anonymous reviewers for their comments and suggestions, which lead to improvements to the paper and its present shape.

Author contributions

All the authors have equally contributed to the final manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

Data sharing does not apply to this article as no data sets were generated or analyzed during the current study.

Table 5. Unique partition codes for the vertices present in first-fifth layer of . $S_5.$

Table 6. Unique partition codes for the vertices present in the $i^{th}$ cycle vertices in $PC{N}_5(y).$

Table 11. Unique edge partition codes for first-fifth layer edges joined due to the vertices in . $S_5.$

Table 12. Unique edge partition codes for $i^{th}$ layer edges joined due to the vertices in $i^{th}$ layer of $S_j$ ($6\le j\le y).$

Table 13. Unique edge partition codes for edges connecting $i^{th}$ cycle to $(i+1{)}^{th}$ ($2\le i\le y-1$) cycle in $PC{N}_5(y).$