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We developed a method to construct all spanning trees for recursive graphs. This approach allows analytical solutions for network properties and optimization problems, enhancing recursive graphs as complex network models.

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Area of Science:

  • Graph theory
  • Network science
  • Statistical physics

Background:

  • Recursive graphs, like the Dorogovtsev-Goltsev-Mendes (DGM) net, are increasingly used to model complex real-world networks.
  • Understanding the properties of spanning trees within these networks is crucial for analyzing network behavior and optimizing their structure.
  • Existing methods may not efficiently capture the full ensemble of spanning trees for recursively generated graphs.

Purpose of the Study:

  • To introduce a novel link-by-link rule-based method for constructing the complete ensemble of spanning trees for recursive graphs.
  • To demonstrate how this method enables exact analytical solutions for large-scale properties of spanning tree ensembles.
  • To show the application of these rules in selecting specific spanning trees with desired properties for network optimization.

Main Methods:

  • A link-by-link rule-based construction algorithm for generating all spanning trees of recursive graphs.
  • Analytical techniques to solve for large-scale properties of the ensemble of spanning trees.
  • Application of growth rules to select subsets of spanning trees with targeted characteristics (e.g., small-world, extended diameter, specific degree distributions).

Main Results:

  • The method successfully constructs all members of the spanning tree ensemble for recursively generated graphs.
  • Exact analytical solutions were obtained for various large-scale properties of these ensembles.
  • The rule-based selection process effectively identifies spanning trees relevant to optimization problems on networks.

Conclusions:

  • The developed method provides a powerful tool for analyzing the ensemble of spanning trees in recursive graphs.
  • This work enhances the utility of recursive graphs as sophisticated models for complex networks.
  • The findings offer new approaches to network analysis and optimization problems by leveraging the structure of spanning trees.