Ratio of Symmetries Between any two n-Node Graphs

TL;DR

This paper establishes a mathematical relationship between the ratios of automorphism group sizes of two graphs and the automorphism orbits of their symmetric difference, revealing a new connection between graph symmetries.

Contribution

It introduces a novel theorem linking the ratios of automorphism groups of two graphs to the automorphism orbits of their symmetric difference.

Findings

Automorphism group size ratios are equal to orbit size ratios.

Provides a new perspective on comparing graph symmetries.

Establishes a formal link between graph differences and symmetries.

Abstract

Given any two graphs on the same vertex set, $G_1 = (V, E_1)$ and $G_2 = (V, E_2)$, along with the difference between the two graphs $\Delta = (E_1 \setminus E_2) \cup (E_2 \setminus E_1)$, we prove that the ratio of the sizes of the two graphs' automorphism groups is equivalent to the ratio of the sizes of $\Delta$'s automorphism orbits in $G_1$ and $G_2$ respectively. This result provides a link between graphs' symmetries that might otherwise seem to be unrelated.