On the satisfaction frequency of spectral characterization conditions

TL;DR

This paper develops a theoretical framework using algebraic and probabilistic methods to estimate how often graphs meet spectral characterization conditions, providing initial conjectures.

Contribution

It introduces a novel algebraic-random matrix approach to analyze the frequency of spectral characterization conditions in graphs.

Findings

Formulated conjectures on the probability of graphs satisfying spectral conditions.

Rephrased spectral conditions in terms of Z[x]-modules and studied their distribution.

Applied the framework to conditions involving the walk matrix determinant and characteristic polynomial discriminant.

Abstract

We give the first specific conjectures on how frequently graphs satisfy sufficient conditions for being uniquely characterized by spectral information. These conjectures arise from a theoretical framework that we developed based on abstract-algebraic random matrix statistics. Specifically, we rephrase conditions from the literature in terms of Z[x]-modules associated to the adjacency matrix, and study the distribution of those modules in analytically tractable profinite random matrix ensembles. We applied this new method to two distinct conditions. The first requires square-freeness of the determinant of the walk matrix, and the second uses the discriminant of the characteristic polynomial.