Isospectral discrete and quantum graphs with the same flip counts and nodal counts

Abstract

The existence of non-isomorphic graphs which share the same Laplace spectrum (to be referred to as isospectral graphs) leads naturally to the following question: What additional information is required in order to resolve isospectral graphs? It was suggested by Band, Shapira and Smilansky that this might be achieved by either counting the number of nodal domains or the number of times the eigenfunctions change sign (the so-called flip count) [1,2]. Recently examples of (discrete) isospectral graphs with the same flip count and nodal count have been constructed by K. Ammann by utilising Godsil-McKay switching [3]. Here we provide a simple alternative mechanism that produces systematic examples of both discrete and quantum isospectral graphs with the same flip and nodal counts. ‡ The term 'cospectral' is used in place of 'isospectral' by some other authors. arXiv:1801.05246v1 [math-ph]

Neumann Domains on Quantum Graphs

“…It was conjectured that such graphs would have different nodal count [27], or in other words that nodal count resolves isospectrality 4 . This conjecture in its most general form have been refuted by now in [13,32,29] (and in [21] for manifolds). Yet, given the discussion above, one may ask whether isospectrality is resolved by combining both the nodal count and the Neumann count.…”

Topologically induced spectral behavior: the example of quantum graphs

“…As is the PDE case, the answer is negative and one can find classes of isospectral graphs [9]. The topology plays again a role here, for instance through the relation between the nodal count of the graph Laplacian with Kirchhoff coupling at the vertices and the first Betti number of the corresponding graph [2], see also [6,57]. • various problems concern the spectral optimization with respect to graph properties.…”