# Resolvability of Hamming Graphs

## TL;DR

This paper characterizes the resolvability of Hamming graphs and hypercubes, introduces efficient methods to determine resolving sets, and demonstrates practical applications in protein sequence analysis using low-dimensional vector representations.

## Contribution

It provides a new linear system characterization of Hamming graph resolvability and develops ILP and Gr"obner basis methods for assessing resolving sets.

## Key findings

- Characterized resolvability of Hamming graphs via linear systems.
- Developed ILP and Gr"obner basis methods for resolving set determination.
- Identified a resolving set of size 77 in octapeptide sequences.

## Abstract

A subset of vertices in a graph is called resolving when the geodesic distances to those vertices uniquely distinguish every vertex in the graph. Here, we characterize the resolvability of Hamming graphs in terms of a constrained linear system and deduce a novel but straightforward characterization of resolvability for hypercubes. We propose an integer linear programming method to assess resolvability rapidly, and provide a more costly but definite method based on Gr\"obner bases to determine whether or not a set of vertices resolves an arbitrary Hamming graph. As proof of concept, we identify a resolving set of size 77 in the metric space of all octapeptides (i.e., proteins composed of eight amino acids) with respect to the Hamming distance; in particular, any octamer may be readily represented as a 77-dimensional real-vector. Representing k-mers as low-dimensional numerical vectors may enable new applications of machine learning algorithms to symbolic sequences.