@MISC{Sharp_degeneracyin, author = {Richard Sharp}, title = {DEGENERACY IN THE LENGTH SPECTRUM FOR METRIC GRAPHS}, year = {} }

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Abstract

Abstract. In this note we show that the length spectrum for metric graphs exhibit a very high degree of degeneracy. More precisely, we obtain an asymptotic for the number of pairs of closed geodesic (or closed cycles) with the same metric length. Let G = (V, E) be a finite graph with vertices V and edges E. We write E o for the set of oriented edges; for e ∈ E o, ē ∈ E o denotes the same unoriented edge with orientation reversed. (In the physics literature, the vertices are referred to as nodes and the edges as bonds.) The degree of a vertex is the number of outgoing