Spanning Trees of Recursive Scale-Free Graphs

TL;DR

This paper introduces a recursive, rule-based method to construct and analyze all spanning trees of finitely articulated, recursive graphs, enabling exact solutions for large-scale properties and optimization of specific spanning tree subsets.

Contribution

It provides a novel link-by-link construction method for spanning trees in recursive graphs, allowing exact analysis and targeted subset selection based on desired properties.

Findings

Exact solutions for large-scale spanning tree properties

Method for selecting spanning trees with specific features

Enhanced modeling of complex networks using recursive graphs

Abstract

We present a link-by-link rule-based method for constructing all members of the ensemble of spanning trees for any recursively generated, finitely articulated graph, such as the DGM net. The recursions allow for many large-scale properties of the ensemble of spanning trees to be analytically solved exactly. We show how a judicious application of the prescribed growth rules selects for certain subsets of the spanning trees with particular desired properties (small-world or extended diameter, degree distribution, etc.), and thus provides solutions to several optimization problems. The analysis of spanning trees enhances the usefulness of recursive graphs as sophisticated models for everyday life complex networks.