The Laplacian spectra of the RG-join weighted graphs and related asymptotic network indices

Da Huang; Xing Chen; Cheng Yan; Zhiyong Yu; Da Huang; Xing Chen; Cheng Yan; Zhiyong Yu

doi:10.3934/math.2025810

AIMS Mathematics

2025, Volume 10, Issue 8: 18183-18194. doi: 10.3934/math.2025810

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Theory article Topical Sections

The Laplacian spectra of the RG-join weighted graphs and related asymptotic network indices

1.
School of Mathematics and Physics, Xinjiang Institute of Engineering, Urumqi 830023, Xinjiang, China
2.
College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, China
3.
College of Mathematics and System Science, Xinjiang University, Urumqi 830017, Xinjiang, China

Received: 01 May 2025 Revised: 24 July 2025 Accepted: 01 August 2025 Published: 12 August 2025
MSC : 15A18, 93A16, 93B60

In this article, the Laplacian spectra of the RG-join weighted graphs and the related network indices named network coherence, kirchhoff index, and Laplacian-energy-like invariant are studied via algebraic graph theory and analysis approach. First, the Laplacian spectrum of the weighted RG-join graph is derived, then the union of graphs together with the RG-join operation are applied to form the weighted RG-join graphs with several classic substructures, and the mathematical characterizations for the indices are derived by the L-spectra. In addition, the related asymptotic results are also derived. It is found that, based on the RG-join weighted structure, when the cardinalities of all copy sets inner $ G_2 $ are large enough, the network coherence and the Kirchhoff index will be irrelevant with the quantity of copies in $ G_2 $, and also irrelevant with the edge weight $ d_1 $ in the other subgraph $ G_1 $.
- networked system,
- topological index,
- weighted graph,
- RG-join,
- Laplacian spectra,
- Kirchhoff index
Citation: Da Huang, Xing Chen, Cheng Yan, Zhiyong Yu. The Laplacian spectra of the RG-join weighted graphs and related asymptotic network indices[J]. AIMS Mathematics, 2025, 10(8): 18183-18194. doi: 10.3934/math.2025810

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Abstract

In this article, the Laplacian spectra of the RG-join weighted graphs and the related network indices named network coherence, kirchhoff index, and Laplacian-energy-like invariant are studied via algebraic graph theory and analysis approach. First, the Laplacian spectrum of the weighted RG-join graph is derived, then the union of graphs together with the RG-join operation are applied to form the weighted RG-join graphs with several classic substructures, and the mathematical characterizations for the indices are derived by the L-spectra. In addition, the related asymptotic results are also derived. It is found that, based on the RG-join weighted structure, when the cardinalities of all copy sets inner $ G_2 $ are large enough, the network coherence and the Kirchhoff index will be irrelevant with the quantity of copies in $ G_2 $, and also irrelevant with the edge weight $ d_1 $ in the other subgraph $ G_1 $.

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