Graph-geometric invariants for molecular structures
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The shortest-path distance on graphs might naturally be imagined to be implicated in some sort of innate geometric structure on graphs, such as are oft used to represent molecular structures. In fact it is noted here that in addition to this shortest-path metric, several other metrics may be naturally defined on graphs, sometimes there being more formal similarities with the Euclidean metric, such as can also be associated with a graph once it is embedded in a Euclidean space as is ultimately the case with molecular structures. Five natural purely graph-theoretic metrics are briefly discussed and are utilized in conjunction with several novel graph invariants defined in correspondence with classical geometric invariants for structures in Euclidean space. These graph invariants thereby attain potentially useful geometrical interpretations in terms of curvatures, torsions, and volumina. Such so-interpretable "graph-geometric" invariants then are evaluated for graphs in several different types of sequences, and the results are compared with one another and the corresponding Euclidean-geometric quantities for simple embeddings of these graphs in three-dimensional Euclidean space. It is thence suggested that certain metrics (especially our "quasi-Euclidean" and "resistance-distance" metrics) form a more suitable foundation for a "graph geometry".