Yves Colin de Verdière's work non the $\mu-$invariant of graphs was motivated by discretizing Schrödinger operators on surfaces (endowed with Riemannian metrics). His papers (mainly in French but probably largely readable by somebody speaking English) on the subject might therefore be interesting to you, see for example his book "Spectres de graphe", Cours Spécialisés, 4. Société Mathématique de France, Paris, 1998. viii+114 pp.
Roland Bacher
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