The Cartesian Product and Join Graphs on Edge-Version Atom-Bond Connectivity and Geometric Arithmetic Indices
Xiujun Zhang1, Huiqin Jiang2, Jia-Bao Liu3
1Key Laboratory of Pattern Recognition and Intelligent Information Processing, Institutions of Higher Education of Sichuan Province, Chengdu University, Chengdu 610106, China. woodszhang@cdu.edu.cn.
Molecules (Basel, Switzerland)
|July 18, 2018
Summary
This study calculates the edge atom-bond connectivity (ABCe) and geometric arithmetic (GAe) indices for Cartesian product and join graphs. These findings extend existing graph theory results.
Area of Science:
- Graph Theory
- Chemical Graph Theory
- Discrete Mathematics
Background:
- The Cartesian product and join are fundamental graph operations.
- Atom-bond connectivity (ABCe) and geometric arithmetic (GAe) indices are important molecular descriptors.
- The edge versions of ABCe and GAe indices (ABCe and GAe) are defined based on vertex degrees in line graphs.
Purpose of the Study:
- To compute the ABCe and GAe indices for specific Cartesian product graphs.
- To deduce the ABCe and GAe indices for particular join graphs.
- To expand and refine existing knowledge on graph indices.
Main Methods:
- Utilized the definitions of edge atom-bond connectivity (ABCe) and geometric arithmetic (GAe) indices.
- Applied these definitions to line graphs of Cartesian product graphs (Pn□Pm, Pn□Cm, Pn□Sm).
- Derived the ABCe and GAe indices for various join graphs (Cm+Pn+Sr, Pm+Pn+Pr, Cm+Cn+Cr, Sm+Sn+Sr).
Main Results:
- Obtained exact values for ABCe and GAe indices of Cartesian product graphs Pn□Pm, Pn□Cm, and Pn□Sm.
- Deduced exact values for ABCe and GAe indices of join graphs Cm+Pn+Sr, Pm+Pn+Pr, Cm+Cn+Cr, and Sm+Sn+Sr.
- Presented novel results that enhance and correct previous findings in graph theory.
Conclusions:
- The study successfully computed ABCe and GAe indices for specified classes of Cartesian product and join graphs.
- The derived formulas provide valuable tools for analyzing these graph structures.
- The results contribute to the ongoing development of graph theory and its applications.
Keywords:
Cartesian product graphatom-bond connectivity indexgeometric arithmetic indexjoin graphline graphMore Related Videos
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