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2. Second Eigenvalue Multiplicity

Let $G$ be a graph with maximum degree bounded by $\Delta$. The maximum possible multiplicity of the second eigenvalue of various matrices associated with the graph is of interest. For the following choices of Hermitian matrix $M_G$ associated with $G$, what is the maximum multiplicity of the second largest eigenvalue of $M_G$?

      Cite this as: AimPL: Spectral graph and hypergraph theory: connections and applications, available at http://aimpl.org/spectralhypergraph.