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The purpose of the present manuscript is to collect known results and present some new ones relating to nodal domains on graphs, with special emphasize on nodal counts. Several methods for counting nodal domains will be presented, and their relevance as a tool in spectral analysis will be discussed.
Trace formulae for d-regular graphs are derived and used to express the spectral density in terms of the periodic walks on the graphs under consideration. The trace formulae depend on a parameter w which can be tuned continuously to assign different weights to different periodic orbit contributions. At the special value w = 1, the only periodic orbits which… (More)
We study the distribution of the number of (non-backtracking) periodic walks on large regular graphs. We propose a formula for the ratio between the variance of the number of t-periodic walks and its mean, when the cardinality of the vertex set V and the period t approach ∞ with t/V → τ for any τ. This formula is based on the conjecture that the spectral… (More)
We compute the mean two-point spectral form factor and the spectral number variance for permutation matrices of large order. The two-point correlation function is expressed in terms of generalized divisor functions, which are frequently discussed in number theory. Using classical results from number theory and casting them in a convenient form, we derive… (More)