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Current Organic Synthesis

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ISSN (Print): 1570-1794
ISSN (Online): 1875-6271

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Research Article

Resistance Distance and Kirchhoff Index in Windmill Graphs

Author(s): Muhammad Shoaib Sardar* and Shou-Jun Xu*

Volume 22, Issue 2, 2025

Published on: 04 July, 2024

Page: [159 - 168] Pages: 10

DOI: 10.2174/0115701794299562240606054510

Price: $65

TIMBC 2025
Abstract

Introduction: The objective of this study is to compute the Kirchhoff index and resistance distance for two classes of windmill graphs, namely the French windmill graph and the Dutch windmill graph.

Methods: In this study, G is considered a simple connected graph with vertex set V (G) and edge set E(G). N is supposed to represent a network derived from G by substituting a 1-ohm resistor for each edge of G. In that case, the resistance between υ,ν ∈ V (G) is considered analogous to the resistance between two equivalent nodes in network N. We employed techniques from electrical network theory to compute the resistance distance and Kirchhoff index.

Results: The Kirchhoff index of G is the sum of the resistance distances between all pairs of vertices in G. Our computations revealed specific patterns and relationships in the resistance distances and Kirchhoff indices across different classes of windmill graphs.

Conclusion: In addition, the Kirchhoff index and resistance distance are computed in this study for specific generalizations of these graphs. The derived equations can inspire further investigation into the resistance distance and Kirchhoff index in real-world windmill networks. Additionally, they offer a chemical framework for future research, aiding in the determination of molecular structures and characteristics.

Keywords: Distance, resistance distance, network, star-mesh transformation, kirchhoff index, windmill graphs.

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Graphical Abstract

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