SIAM Journal on Discrete Mathematics, Volume 34, Issue 4, Page 2063-2081, January 2020.
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 Gröbner 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.

中文翻译：

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汉明图的可分辨性

SIAM离散数学杂志，第34卷，第4期，第2063-2081页，2020年1月。
当到顶点的测地线距离唯一地区分图中的每个顶点时，该图中顶点的子集称为解析。在这里，我们根据受约束的线性系统刻画了汉明图的可分辨性，并得出了超立方体可分辨性的新颖而直接的刻画。我们提出一种整数线性规划方法来快速评估可分辨性，并提供一种基于Gröbner基的成本更高但更确定的方法，以确定一组顶点是否可以解析任意汉明图。作为概念证明，我们确定了所有八肽（即由八个氨基酸组成的蛋白质）相对于汉明距离的度量空间中的大小为77的解析集。特别地，任何八聚体都可以容易地表示为77维实向量。