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Both the signed ($D-A$) and unsigned ($D+A$) Laplacian are of interest in spectral graph theory, see eg Cvetkovic: Bibliography on the signless Laplacian eigenvalues: first one hundred references.

Considering the spectral function $D+\rho A$ over the interval $\rho \in [-1,1]$ rather than just the extreme values $\rho={-1,1}$, results in the curves $\lambda_i(\rho)$.

For example, the following fractional spectra are are shown as point interpolations (as opposed to curve following, as in bifurcation theory) corresponding to some random Bernoulli graphs with increasing connectivity:

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These experimental results raise basic questions:

  • If graphs are $M$-cospectral on $\rho = {-1,1}$ are they cospectral for $\rho \in [-1,1]$?

  • Are there no intersections of $\lambda_i(\rho)$ outside $\rho \in [-1,1]$?

  • Does $\lim_{\lambda_i \to \infty}\frac{d\lambda_i}{d\rho} = const?$

  • What's the combinatorial interpretation of intersections in $\rho \in [-1,1]$?

Feel free to add your own conjectures.

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