The Number of Nodal Domains on Quantum Graphs as a Stability Index of Graph Partitions

The Courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of Laplacian-type operators. In particular, it holds for generic eigenfunctions of a quantum graph. The theorem stipulates that, after ordering the eigenvalues as a non decreasing sequence, the number of nodal domains ν _n of the n ^th eigenfunction satisfies n ≥ ν _n. Here, we provide a new interpretation for the Courant nodal deficiency d _n = n - ν _n in the case of quantum graphs. It equals the Morse index - at a critical point - of an energy functional on a suitably defined space of graph partitions. Thus, the nodal deficiency assumes a previously unknown and profound meaning - it is the number of unstable directions in the vicinity of the critical point corresponding to the n ^th eigenfunction. To demonstrate this connection, the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail.