Reducible Iterated Graph Systems: multiscale-freeness and multifractals

TL;DR

This paper extends the theory of Iterated Graph Systems to reducible cases, defining multifractality and multiscale-freeness for fractal graphs, and analyzing their spectra and conditions.

Contribution

It introduces a rigorous framework for reducible Edge IGS, establishing conditions and spectra related to multifractality and multiscale-freeness.

Findings

Both spectra are finite and discrete.

Conditions for multifractality and multiscale-freeness are established.

Fills theoretical gaps in primitive Edge IGS studies.

Abstract

Iterated Graph Systems (IGS) transplant ideas from fractal geometry into graph theory. Building on this framework, we extend Edge IGS from the primitive to the reducible setting. Within this broader context, we formulate rigorous definitions of multifractality and multiscale-freeness for fractal graphs, and we establish conditions that are equivalent to the occurrence of these two phenomena. We further determine the corresponding fractal and degree spectra, proving that both are finite and discrete. These results complete the foundational theory of Edge IGS by filling the gap left by the primitive case studied in [1, 2].