An improved upper bound measure of star complexity of graphs

TL;DR

This paper introduces a tighter upper bound measure for star complexity of graphs using the ABC package, leading to a more linear relationship with information-based complexity in large Erdős-Rényi graphs.

Contribution

The paper develops a new ABC-based upper bound for star complexity, improving the accuracy of complexity measurement in graphs.

Findings

New measure shows a more linear relationship with information-based complexity.

Applied to 1000 Erdős-Rényi graphs with 500 vertices each.

Tighter upper bound improves understanding of graph complexity relationships.

Abstract

In \cite{Standish25c}, I explored the connection between star complexity and information based complexity. Because of the numerical difficulty in computing star complexity, I introduced a proxy measure that is an upper bound to star complexity, and showed a strong albeit non-linear relationship between the measures. In this paper, I introduce a tighter upper bound, by exploiting the well-known ABC package used to optimise logic circuits. In testing the new measure, I found that I had been computing the {\em formula complexity} variant of star complexity, rather than the tighter {\em circuit complexity} variant. Since Jukna clearly states the connection between star complexity and circuit complexity, I have modified the graph walking algorithm to capture circuit complexity rather than formula complexity. With this new ABC-based measure, applied to a set of 1000 500 vertex Erd\"os-Renyi graphs, a more linear relationship between star complexity and information based complexity is found.