Descriptive complexity of controllable graphs

TL;DR

This paper studies controllable graphs, characterizes their isomorphism problem through various mathematical lenses, and provides an efficient algorithm for graph isomorphism in most cases.

Contribution

It introduces a new controllability concept for graphs, links graph isomorphism to other problems, and offers a polynomial-time algorithm applicable to almost all graphs.

Findings

Characterization of controllable graphs via combinatorial, geometric, and logical problems.

Polynomial-time algorithm for graph isomorphism for almost all graphs.

Insights into the structure and symmetry of controllable graphs.

Abstract

Let $G$ be a graph on $n$ vertices with adjacency matrix $A$, and let $\mathbf{1}$ be the all-ones vector. We call $G$ controllable if the set of vectors $\mathbf{1}, A\mathbf{1}, \dots, A^{n-1}\mathbf{1}$ spans the whole space $\mathbb{R}^n$. We characterize the isomorphism problem of controllable graphs in terms of other combinatorial, geometric and logical problems. We also describe a polynomial time algorithm for graph isomorphism that works for almost all graphs.