Abstract
The Sturm–Liouville equations on the edges of a metric connected graph together with the boundary and matching conditions at the vertices generate a spectral problem for a self-adjoint operator. It is shown that if the graph is not cyclically connected, then the maximal multiplicity of an eigenvalue of such an operator is μ + gT − pTin, where μ is the cyclomatic number of the graph, and gT and pTin are the number of edges and the number of interior vertices, respectively, for the tree obtained by contracting all the cycles of the graph into vertices. If the graph is cyclically connected, then the maximal multiplicity of an eigenvalue is μ + 1.
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