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arXiv · 1907.05974

Resolvability of Hamming Graphs

Lucas Laird·Richard C. Tillquist·Stephen Becker·Manuel E. Lladser
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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.

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Lucas Laird, Richard C. Tillquist, Stephen Becker, Manuel E. Lladser. 2019-07-12. Resolvability of Hamming Graphs. https://doi.org/10.1137/19m1274511

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