Isospectral graphs with identical nodal counts

Abstract

According to a recent conjecture, isospectral objects have different nodal count sequences. We study generalized Laplacians on discrete graphs, and use them to construct the first non-trivial counter-examples to this conjecture. In addition, these examples demonstrate a surprising connection between isospectral discrete and quantum graphs.

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• Quantum graphs -- Generic eigenfunctions and their nodal count and Neumann count statistics
• Isospectral discrete and quantum graphs with the same flip counts and nodal counts
• Neumann Domains on Quantum Graphs
• Role of Iso-connectivity Topologies in Multi-agent Interactions
• Changing gears: Isospectrality via eigenderivative transplantation