Yes, Brendan McKay showed that almost all trees have mates that are simultaneously cospectral in both adjacency and Laplacian spectra. And more.
http://users.cecs.anu.edu.au/~bdm/papers/SpectralTrees.pdf
Edit: I wondered briefly what the smallest pair of such trees would be, and a few minutes of Sage told me that there are two on 11 vertices.
And here they are (for some reason I am having difficulty with the image uploader):
Yes, Brendan McKay showed that almost all trees have mates that are simultaneously cospectral in both adjacency and Laplacian spectra. And more.
http://users.cecs.anu.edu.au/~bdm/papers/SpectralTrees.pdf
Edit: I wondered briefly what the smallest pair of such trees would be, and a few minutes of Sage told me that there are two on 11 vertices.
Yes, Brendan McKay showed that almost all trees have mates that are simultaneously cospectral in both adjacency and Laplacian spectra. And more.
http://users.cecs.anu.edu.au/~bdm/papers/SpectralTrees.pdf
Edit: I wondered briefly what the smallest pair of such trees would be, and a few minutes of Sage told me that there are two on 11 vertices.
And here they are (for some reason I am having difficulty with the image uploader):
