  Using an exhaustive search on cubic graphs of order 16, we find a unique
cospectral pair with different chromatic indexes. This example indicates that
the chromatic index of a regular graph is not characterized by its spectrum,
which answers a question recently posed in [O. Etesami, W. H. Haemers, On
NP-hard graph properties characterized by the spectrum, Discrete Appl. Math.,
285(2020)526-529]. We prove that any orthogonal matrix representing the
similarity between the two adjacency matrices of the cospectral pair cannot be
rational. This implies that the cospectral pair cannot be obtained using the
original GM-switching method or its generalizations based on rational
orthogonal matrices.
